Dynamic Facility Location Problem Research Articles

The dynamic facility location problem with mobile production units: A waste recycling application

This paper presents a multi-period location–production problem involving small recycling units that are embedded in standard containers and can, therefore, be relocated from site to site at short notice. The waste generated over time at different locations can be stored up to a specific limit, at which point recycling must occur. A mixed-integer program is implemented to plan the relocation of the mobile recycling units in such a way that the total cost resulting from waste transport and facility relocation is minimized. To solve large instances, three heuristics are developed that form the basis for a detailed computational study. The results reveal that mobile recycling units can significantly reduce the total costs compared to centralized recycling because (a) a larger number of mobile recycling units, (b) larger storage capacities, and (c) more short-term relocation possibilities increase the optimization scope and thus the possibility to reduce the total transport distances, and thereby, the costs. However, achieving this requires intelligent planning that considers the complex interdependencies between the influencing parameters and balances the existing trade-offs.

A Dynamic Facility Location Problem with Random Demand and the Restoration of Transportation Networks in Disaster Relief

A Dynamic Facility Location Problem with Random Demand and the Restoration of Transportation Networks in Disaster Relief

Online Facility Location in Evolving Metrics

The Dynamic Facility Location problem is a generalization of the classic Facility Location problem, in which the distance metric between clients and facilities changes over time. Such metrics that develop as a function of time are usually called “evolving metrics”, thus Dynamic Facility Location can be alternatively interpreted as a Facility Location problem in evolving metrics. The objective in this time-dependent variant is to balance the trade-off between optimizing the classic objective function and the stability of the solution, which is modeled by charging a switching cost when a client’s assignment changes from one facility to another. In this paper, we study the online variant of Dynamic Facility Location. We present a randomized O(logm+logn)-competitive algorithm, where m is the number of facilities and n is the number of clients. In the first step, our algorithm produces a fractional solution, in each timestep, to the objective of Dynamic Facility Location involving a regularization function. This step is an adaptation of the generic algorithm proposed by Buchbinder et al. in their work “Competitive Analysis via Regularization.” Then, our algorithm rounds the fractional solution of this timestep to an integral one with the use of exponential clocks. We complement our result by proving a lower bound of Ω(m) for deterministic algorithms and lower bound of Ω(logm) for randomized algorithms. To the best of our knowledge, these are the first results for the online variant of the Dynamic Facility Location problem.

3-D Dynamic UAV Base Station Location Problem

We address a dynamic covering location problem of an unmanned aerial vehicle base station (UAV-BS), in which the location sequence of a single UAV-BS in a wireless communication network is determined to satisfy data demand arising from ground users. This problem is especially relevant in the context of smart grid and disaster relief. The vertical movement ability of the UAV-BS and nonconvex covering functions in wireless communication restrict utilizing classical planar covering location approaches. Therefore, we develop new formulations to this emerging problem for a finite time horizon to maximize the total coverage. In particular, we develop a mixed-integer nonlinear programming formulation that is nonconvex in nature and propose a Lagrangean decomposition algorithm (LDA) to solve this formulation. Because of the high complexity of the problem, the LDA is still unable to find good local solutions to large-scale problems. Therefore, we develop a continuum approximation (CA) model and show that CA would be a promising approach in terms of both computational time and solution accuracy. Our numerical study also shows that the CA model can be a remedy to build efficient initial solutions for exact solution algorithms. Summary of Contribution: This paper addresses a facet of mixed integer nonlinear programming formulations. Dynamic facility location problems (DFLPs) arise in a wide range of applications. However, classical DFLPs typically focus on the two-dimensional spaces. Emerging technologies in wireless communication and some other promising application areas, such as smart grids, have brought new location problems that cannot be solved with classical approaches. For practical reasons, many research attempts to solve this new problem, especially by researchers whose primary research area is not OR, have seemed far from analyzing the characteristics of the formulations. Rather, solution-oriented greedy heuristics have been proposed. This paper has two main objectives: (i) to close the gap between practical and theoretical sides of this new problem with the help of current knowledge that OR possesses to solve facility location problems and (ii) to support the findings with an exhaustive computational study to show how these findings can be applied to practice.

Heuristics for the dynamic facility location problem with modular capacities

This paper studies the Dynamic Facility Location Problem with Modular Capacities (DFLPM). It generalizes several facility location problems and consists in determining locations and sizes of facilities to minimize location and demand allocation costs with decisions taken periodically over a planning horizon. The DFLPM is solved using heuristics tailored for different scenarios and cost structures. We propose three linear relaxation based heuristics (LRH) and an evolutionary heuristic that hybridizes a genetic algorithm with a variable neighborhood descent (GA+VND). We adapt benchmark instances from the literature to yield several representations of scenarios and parameters structures. Experiments are reported comparing the heuristics to a state-of-the-art mixed integer programming (MIP) formulation for the problem. We show that the performance of the methods depends on the characteristics of the instance solved. For the benchmark instances, the LRH improved by VND finds solutions within 0.02% of the optimal ones in less than half of the time of the MIP. For the scenarios where construction costs are higher and module sizes are lower, the GA+VND proved to be effective to solve the problem, outperforming the LRH and the MIP. We also discuss the results from a practitioner point of view to identify situations where each method is preferable.

Lagrangian Heuristics for Large-Scale Dynamic Facility Location with Generalized Modular Capacities

We consider the dynamic facility location problem with generalized modular capacities, a multiperiod facility location problem in which the costs for capacity changes may differ for all pairs of capacity levels. The problem embeds a complex cost structure and generalizes several existing facility location problems, such as those that allow temporary facility closing or capacity expansion and reduction. As the model may become very large, general-purpose mixed-integer programming (MIP) solvers are limited to solving instances of small to medium size. In this paper, we extend the generalized model to the case of multiple commodities. We propose Lagrangian heuristics, based on subgradient and bundle methods, to find good quality solutions for large-scale instances with up to 250 facility locations and 1,000 customers. To improve the final solution quality, a restricted MIP model is solved based on the information collected through the solution of the Lagrangian dual. Computational results show that the Lagrangian-based heuristics provide highly reliable results for all problem variants considered. They produce good quality solutions in short computing times even for instances where state-of-the-art MIP solvers do not find feasible solutions. The strength of the formulation also allows the method to provide tight bounds on the optimal value. Data and the online appendix are available at https://doi.org/10.1287/ijoc.2016.0738 .

An approximation algorithm for the dynamic facility location problem with outliers

In this article, we investigate the dynamic (multi-period) facility location problem with potentially unserved clients or outliers. We propose a 3-approximation primal-dual algorithm based on an integer linear program formulation of the problem. We further improve the approximation ratio to 2 by combining the cost scaling and greedy improvement techniques.

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.

A Continuum Approximation Approach to the Dynamic Facility Location Problem in a Growing Market

This paper proposes a continuum approximation (CA) model to solve the dynamic facility location problem for a large-scale growing market. The objective is to determine the optimal facility location and deployment time that minimize the costs for facility construction and customer service in a planning horizon. To overcome computational challenges, the CA model determines the optimal facility density in the spatiotemporal continuum. Then we propose a tube model to discretize the resulting continuous facility density function into a set of time-varying facility location trajectories. To enforce consistency in facility location over time, an iterative regulation procedure based on a penalty method is applied. We present convergence properties of the proposed procedure and further derive conditions under which the CA approach and the tube model yield tight approximation error bounds. We conduct a series of numerical experiments to illustrate the applicability and computational performance (e.g., accuracy and convergence) of the proposed modeling framework, first via comparison with discrete model counterparts using hypothetical data, and then via application to an empirical case for the state of Illinois. Our results show that the proposed method effectively solves the dynamic facility location problem to reasonable accuracy. Various managerial insights are also drawn.

A Benders based rolling horizon algorithm for a dynamic facility location problem

This study presents a well-known capacitated dynamic facility location problem (DFLP) that satisfies the customer demand at a minimum cost by determining the time period for opening, closing, or retaining an existing facility in a given location. To solve this challenging NP-hard problem, this paper develops a unique hybrid solution algorithm that combines a rolling horizon algorithm with an accelerated Benders decomposition algorithm. Extensive computational experiments are performed on benchmark test instances to evaluate the hybrid algorithm’s efficiency and robustness in solving the DFLP problem. Computational results indicate that the hybrid Benders based rolling horizon algorithm consistently offers high quality feasible solutions in a much shorter computational time period than the stand-alone rolling horizon and accelerated Benders decomposition algorithms in the experimental range.

Solving a dynamic facility location problem with partial closing and reopening

Motivated by an industrial application, we consider a recently introduced multi-period facility location problem with multiple commodities and multiple capacity levels. The problem allows for the relocation of facilities, as well as for the temporary closing of parts of the facilities, while other parts remain open. In addition, it uses particular capacity constraints that involve integer rounding of the allocated demands. In this paper, we propose a strong formulation for the problem, as well as a hybrid heuristic that first applies Lagrangian relaxation and then constructs a restricted mixed-integer programming model based on the previously obtained Lagrangian solutions. Computational results for large-scale instances emphasize the usefulness of the heuristic in practice. While general-purpose mixed-integer programming solvers do not find feasible solutions for about half of the instances, the heuristic consistently provides high-quality solutions in short computing times, as well as tight bounds on their optimality.

Dynamic Facility Location Problem Considering Age of Facility and Conversion

Dynamic Facility Location Problem Considering Age of Facility and Conversion

Dynamic Facility Location with Generalized Modular Capacities

Location decisions are frequently subject to dynamic aspects such as changes in customer demand. Often, flexibility regarding the geographic location of facilities, as well as their capacities, is the only solution to such issues. Even when demand can be forecast, finding the optimal schedule for the deployment and dynamic adjustment of capacities remains a challenge, especially when the cost structure for these adjustments is complex. In this paper, we introduce a unifying model that generalizes existing formulations for several dynamic facility location problems and provides stronger linear programming relaxations than the specialized formulations. In addition, the model can address facility location problems where the costs for capacity changes are defined for all pairs of capacity levels. To the best of our knowledge, this problem has not been addressed in the literature. We apply our model to special cases of the problem with capacity expansion and reduction or temporary facility closing and reopening. We prove dominance relationships between our formulation and existing models for the special cases. Computational experiments on a large set of randomly generated instances with up to 100 facility locations and 1,000 customers show that our model can obtain optimal solutions in shorter computing times than the existing specialized formulations.

An efficient variable neighborhood search for solving a robust dynamic facility location problem in emergency service network

In this study, we propose a robust variant of a dynamic facility location problem that arises from optimizing the emergency service network of Police Special Forces Units (PSFUs) in the Republic of Serbia. We present for the first time a mathematical programming formulation of the problem under consideration. We further propose a Variable Neighborhood Search (VNS) method with an efficient local search procedure for solving real-life problem instances that remained out of reach of CPLEX solver. The results presented in this paper may help in optimizing the network of PSFUs and other security networks as well.

An approximation algorithm for the dynamic facility location problem with submodular penalties

In this paper, we study the dynamic facility location problem with submodular penalties (DFLPSP). We present a combinatorial primal-dual 3-approximation algorithm for the DFLPSP.

Simple dynamic location problem with uncertainty: a primal-dual heuristic approach

In this paper, the simple dynamic facility location problem is extended to uncertain realizations of the potential locations for facilities and the existence of customers as well as fixed and variable costs. With limited knowledge about the future, a finite and discrete set of scenarios is considered. The decisions to be made are where and when to locate the facilities, and how to assign the existing customers over the whole planning horizon and under each scenario, in order to minimize the expected total costs. Whilst assignment decisions can be scenario dependent, location decisions have to take into account all possible scenarios and cannot be changed according to each scenario in particular. We first propose a mixed linear programming formulation for this problem and then we present a primal-dual heuristic approach to solve it. The heuristic was tested over a set of randomly generated test problems. The computational results are provided.

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.]

Solving dynamic optimisation problems with revolutionary algorithms

The optimisation of dynamic problems is both widespread and difficult. When conducting dynamic optimisation, a balance between re-initialisation and computational expense has to be found. There are multiple approaches to this. In parallel genetic algorithms, multiple sub-populations concurrently try to optimise a potentially dynamic problem. But as the number of sub-population increases, their efficiency decreases. Cultural algorithms provide a framework that has the potential to make optimisations more efficient. But they adapt slowly to changing environments. We thus suggest a confluence of these approaches: revolutionary algorithms. These algorithms seek to extend the evolutionary and cultural aspects of the former two approaches with a notion of the political. By modelling how belief systems are changed by means of revolution, these algorithms provide a framework to model and optimise dynamic problems in an efficient fashion. The superiority of revolutionary algorithms over cultural and purely genetic algorithms is demonstrated in the solving of a standard dynamic facility location problem.

Approximating soft-capacitated facility location problem with uncertainty

In this paper we devise the stochastic and robust approaches to study the soft-capacitated facility location problem with uncertainty. We first present a new stochastic soft-capacitated model called The 2-Stage Soft Capacitated Facility Location Problem and solve it via an approximation algorithm by reducing it to linear-cost version of 2-stage facility location problem and dynamic facility location problem. We then present a novel robust model of soft-capacitated facility location, The Robust Soft Capacitated Facility Location Problem. To solve it, we improve the approximation algorithm proposed by Byrka et al. (LP-rounding algorithms for facility-location problems. CoRR, 2010a) for RFTFL and then treat it similarly as in the stochastic case. The improvement results in an approximation factor of $$\alpha + 4$$ for the robust fault-tolerant facility location problem, which is best so far.

Optimization models for the dynamic facility location and allocation problem

The design of logistic distribution systems is one of the most critical and strategic issues in industrial facility management. The aim of this study is to develop and apply innovative mixed integer programming optimization models to design and manage dynamic (i.e. multi-period) multi-stage and multi-commodity location allocation problems (LAP). LAP belong to the NP-hard complexity class of decision problems, and the generic occurrence requires simultaneous determination of the number of logistic facilities (e.g. production plants, warehousing systems, distribution centres), their locations, and assignment of customer demand to them. The proposed models use a mixed integer linear programming solver to find solutions in complex industrial applications even when several entities are involved (production plants, distribution centres, customers, etc.). Lastly, the application of the proposed models to a significant case study is presented and discussed.