Discrete Facility Location Problem Research Articles

A congested capacitated location problem with continuous network demand

This paper presents a multi-objective mixed-integer non-linear programming model for a congested multiple-server discrete facility location problem with uniformly distributed demands along the network edges. Regarding the capacity of each facility and the maximum waiting time threshold, the developed model aims to determine the number and locations of established facilities along with their corresponding number of assigned servers such that the traveling distance, the waiting time, the total cost, and the number of lost sales (uncovered customers) are minimized simultaneously. Also, this paper proposes modified versions of some of the existing heuristics and metaheuristic algorithms currently used to solve NP-hard location problems. Here, the memetic algorithm along with its modified version called the stochastic memetic algorithm, as well as the modified add and modified drop heuristics are used as the solution methods. Computational results and comparisons demonstrate that although the results obtained from the developed stochastic memetic algorithm are slightly better, the applied memetic algorithm could be considered as the most efficient approach in finding reasonable solutions with less required CPU times.

A Clustering-based Simulated Annealing Algorithm with Taguchi Method for the Discrete Ordered Median Problem

Researchers have studied discrete location problems for a long time because of their importance in practice. The Discrete Ordered Median Problem (DOMP) generalizes discrete facility location problems. The DOMP generalizes the main facility location problems' objective functions such as the p-median, p-center and p-centdian location problems. As these problems, also known as the problems of location-allocation, have NP-hard structure, it is inevitable to use heuristic methods for solution. In this study, a metaheuristic algorithmic suggestion will be put forward by examining the DOMP to find optimal solutions. For that purpose, we proposed a Simulated Annealing (SA) metaheuristic with K-means Clustering Algorithm in initialization for the DOMP. Novel approaches for initial solution and K-exchange algorithm-based neighborhoods for local search were analysed. In addition, best level of selected parameters were determined by Taguchi method. Forty common p-median instances derived from OR-LIB were used to test the SA performance, and the results were compared with three state-of-art algorithms in the literature. According to the computational results, 21 best solutions were obtained on instances despite gap values and CPU times increasing proportionally to the scale of the instances. In a conclusion, the proposed clustering-based SA algorithm is competitive and can be a robust alternative for the DOMP.

Neutrosophic Discrete Facility Location Problems

Discrete facility location problems are classified as types of facility location problems, wherein decisions on choosing facilities in specific locations are made to serve the demand points of customers, thus minimizing the total cost. The covering- and median-based problems are the common classified types of discrete facility location problems, which both comprise different classes of discrete problems as reviewed in this research. However, the discrete facility location problems shown in deterministic and known information and data under uncertain, vague, and ambiguous environments have usually been solved using intuitionistic fuzzy approaches. Neutrosophic is recently applied to tackle the uncertainty and ambiguity of information and data. This paper considered solving the discrete facility location problems under the neutrosophic environment, wherein the information of the locations, distances, times, and costs is uncertain. The mathematical models for the main types of neutrosophic discrete facility location problems, which remain unclear till now despite previous related works, are formulated in this study. Numerical examples demonstrated testing of the neutrosophic discrete models and comparison with the optimization solutions obtained from the normal situations.

A competitive facility location problem using distributor Stackelberg game approach in multiple three-level supply chains

This paper studies a discrete facility location problem in two different supply chains including suppliers, distributors and customers. The current and new distribution firm act as the leader and the follower respectively in a Stackelberg game. The model considers the lead-time as a competitive factor between the new Distribution Centres (DCs) and the existing ones as well as optimising the location and number of to-be-established DCs in the new supply chain by minimising the total cost. The Branch and Bound as well as Genetic Algorithm (GA) are used for solving the model, and the results are compared with the situation that there is no competitor results show in case of competition; there is more possibility of lost sales ignoring the rival decision in making strategic decision of locating new DCs. While, when there is no other player, decisions are taken independently as the only possible option for consumers.

Multi-supply multi-capacitated p-median location optimization via a hybrid bi-level intelligent algorithm

The Capacitated P-Median Problem (CPMP) is a classic discrete facility location problem in which p capacitated facilities (medians) are selected from an economic point of view to serve a set of demand vertexes so that the total demand assigned to each of the facilities (medians) does not exceed its capacity. Since the capacity used by each facility is fixed and does not depend on the vertex demand, the traditional capacitated p-median problem’s total capacity is underutilized. To make full use of the total capacity, a more generalized model, namely, the Multi-supply Multi-Capacitated Location Problem (MMCLP), is considered in this paper, which assumes not only that each facility is allowed to open with different capacities while meeting the constraint of total capacity but also that each demand vertex can accept supplies from different facilities. To solve the MMCLP effectively, we propose a hybrid bi-level optimization framework based on decomposition. First, the original MMCLP is decomposed into numerous mixed-integer programming (MIP) sub-problems, i.e., inner optimization is performed, and for outer optimization, a Genetic Algorithm based on Co-Evolution of Population and Neighborhood (GACEPN) is proposed to find better sub-problems near the good sub-problems among all the candidate sub-problems. The inner optimization problem is solved by a commercial MIP solver. The results obtained are returned to the outer layer so that the neighborhood of the outer layer optimization can be adjusted appropriately to generate better sub-problems, thereby approaching the optimal solution of the original problem. The experiments show that for a large-scale MMCLP, compared to the Gurobi solver, the proposed bi-level algorithm (a hybrid of the GACEPN and Gurobi) can find a better solution in less time.

Solving Classic Discrete Facility Location Problems Using Excel Spreadsheets

There are a variety of discrete facility location models that have practical relevance for operations management and management science courses. Integer linear programming (ILP) is the standard technique for solving such problems. An alternative approach that is often conceptually appealing to students is to pose the problem as one of finding the best possible subset of p facilities out of n possible candidates. I developed an Excel workbook that allows students to interactively evaluate the quality of different subsets, to run a VBA macro that finds the optimal subset, or to solve an ILP formulation that finds the optimal subset. Spreadsheets are available for five classic discrete location models: (1) the location set-covering problem, (2) the maximal covering location problem, (3) the p-median problem, (4) the p-centers problem, and (5) the simple plant location problem. The results from an assignment in a master’s-level business analytics course indicate that the workbook facilitates a better conceptual understanding of the precise nature of the discrete facility location problems by showing that they can be solved via enumeration of all possible combinations of p subsets that can be drawn from n candidate locations. More important, students directly observe the superiority of ILP as a solution approach as n increases and as p approaches n/2.

Solution of asymmetric discrete competitive facility location problems using ranking of candidate locations

We address a discrete competitive facility location problem with an asymmetric objective function and a binary customer choice rule. Both an integer linear programming formulation and a heuristic optimization algorithm based on ranking of candidate locations are designed to solve the problem. The proposed population-based heuristic algorithm is specially adapted for the discrete facility location problems by using their features such as geographical distances and the maximal possible utility of candidate locations, which can be evaluated in advance. The performance of the proposed algorithm was experimentally investigated by solving different instances of the model with real data of municipalities in Spain.

Gradual covering location problem with multi-type facilities considering customer preferences

In this paper, we address a discrete facility location problem where a retailer aims at locating new facilities with possibly different characteristics. Customers visit the facilities based on their preferences which are represented as probabilities. These probabilities are determined in a novel way by using a fuzzy clustering algorithm. It is assumed that the sum of the probabilities with which customers at a given demand zone patronize different types of facilities is equal to one. However, among the same type of facilities they choose the closest facility, and the strength at which this facility covers the customer is based on two distances referred to as full coverage distance and gradual (partial) coverage distance. If the distance between the customer location and the closest facility is smaller (larger) than the full (partial) coverage distance, this customer is fully (not) covered, whereas for all distance values between full and partial coverage, the customer is partially covered. Both distance values depend on both the customer attributes and the type of the facility. Furthermore, facilities can only be opened if their revenue exceeds a certain threshold value. A final restriction is incorporated into the model by defining a minimum separation distance between the same facility types. This restriction is also extended to the case where a minimum threshold distance exists among facilities of different types. The objective of the retailer is to find the optimal locations and types of the new facilities in order to maximize its profit. Two versions of the problem are formulated using integer linear programming, which differ according to whether the minimum separation distance applies to the same facility type or different facility types. The resulting integer linear programming models are solved by three approaches: commercial solver CPLEX, heuristics based on Lagrangean relaxation, and local search implemented with 1-Add and 1-Swap moves. Apart from experimentally assessing the accuracy and the efficiency of the solution methods on a set of randomly generated test instances, we also carry out sensitivity analysis using a real-world problem instance.

On multi-criteria chance-constrained capacitated single-source discrete facility location problems

This work aims at investigating multi-criteria modeling frameworks for discrete stochastic facility location problems with single sourcing. We assume that demand is stochastic and also that a service level is imposed. This situation is modeled using a set of probabilistic constraints. We also consider a minimum throughput at the facilities to justify opening them. We investigate two paradigms in terms of multi-criteria optimization: vectorial optimization and goal programming. Additionally, we discuss the joint use of objective functions that are relevant in the context of some humanitarian logistics problems. We apply the general modeling frameworks proposed to the so-called stochastic shelter site location problem. This is a problem emerging in the context of preventive disaster management. We test the models proposed using two real benchmark data sets. The results show that considering uncertainty and multiple objectives in the type of facility location problems investigated leads to solutions that may better support decision making.

Heuristic Solutions to the Facility Location Problem with General Bernoulli Demands

In this paper, a heuristic procedure is proposed for the facility location problem with general Bernoulli demands. This is a discrete facility location problem with stochastic demands that can be formulated as a two-stage stochastic program with recourse. In particular, facility locations and customer assignments must be decided here and now, i.e., before knowing the customers who will actually require to be served. In a second stage, service decisions are made according to the actual requests. The heuristic proposed consists of a greedy randomized adaptive search procedure followed by a path relinking. The heterogeneous Bernoulli demands make prohibitive the computational effort for evaluating feasible solutions. Thus the expected cost of a feasible solution is simulated when necessary. The results of extensive computational tests performed for evaluating the quality of the heuristic are reported, showing that high-quality feasible solutions can be obtained for the problem in fairly small computational times. The online supplement is available at https://doi.org/10.1287/ijoc.2017.0755 .

A polynomial-time algorithm for the discrete facility location problem with limited distances and capacity constraints

The objective in terms of the facility location problem with limited distances is to minimize the sum of distance functions from the facility to its clients, but with a limit on each of these distances, from which the corresponding function becomes constant. The problem is applicable in situations where the service provided by the facility is insensitive after given threshold distances. In this paper, we propose a polynomial-time algorithm for the discrete version of the problem with capacity constraints regarding the number of served clients. These constraints are relevant for introducing quality measures in facility location decision processes as well as for justifying the facility creation.

A bi-objective approach to discrete cost-bottleneck location problems

This paper considers a family of bi-objective discrete facility location problems with a cost objective and a bottleneck objective. A special case is, for instance, a bi-objective version of the (vertex) p-centdian problem. We show that bi-objective facility location problems of this type can be solved efficiently by means of an $$\varepsilon $$ -constraint method that solves at most $$(n-1)\cdot m$$ minisum problems, where n is the number of customer points and m the number of potential facility sites. Additionally, we compare the approach to a lexicographic $$\varepsilon $$ -constrained method that only returns efficient solutions and to a two-phase method relying on the perpendicular search method. We report extensive computational results obtained from several classes of facility location problems. The proposed algorithm compares very favorably to both the lexicographic $$\varepsilon $$ -constrained method and to the two phase method.

An improved Lagrangian relaxation and dual ascent approach to facility location problems

The literature knows semi-Lagrangian relaxation as a particular way of applying Lagrangian relaxation to certain linear mixed integer programs such that no duality gap results. The resulting Lagrangian subproblem usually can substantially be reduced in size. The method may thus be more efficient in finding an optimal solution to a mixed integer program than a “solver” applied to the initial MIP formulation, provided that “small” optimal multiplier values can be found in a few iterations. Recently, a simplification of the semi-Lagrangian relaxation scheme has been suggested in the literature. This “simplified” approach is actually to apply ordinary Lagrangian relaxation to a reformulated problem and still does not show a duality gap, but the Lagrangian dual reduces to a one-dimensional optimization problem. The expense of this simplification is, however, that the Lagrangian subproblem usually can not be reduced to the same extent as in the case of ordinary semi-Lagrangian relaxation. Hence, an effective method for optimizing the Lagrangian dual function is of utmost importance for obtaining a computational advantage from the simplified Lagrangian dual function. In this paper, we suggest a new dual ascent method for optimizing both the semi-Lagrangian dual function as well as its simplified form for the case of a generic discrete facility location problem and apply the method to the uncapacitated facility location problem. Our computational results show that the method generally only requires a very few iterations for computing optimal multipliers. Moreover, we give an interesting economic interpretation of the semi-Lagrangian multiplier(s).

A Survey of Discrete Facility Location Problems

A Survey of Discrete Facility Location Problems

Threshold robustness in discrete facility location problems: a bi-objective approach

The two best studied facility location problems are the \(p\)-median problem and the uncapacitated facility location problem (Daskin, Network and discrete location: models, algorithms, and applications. Wiley, New York, 1995; Mirchandani and Francis, Discrete location theory. Wiley, New York, 1990). Both seek the location of the facilities minimizing the total cost, assuming no uncertainty in costs exists, and thus all parameters are known. In most real-world location problems the demand is not certain, because it is a long-term planning decision, and thus, together with the minimization of costs, optimizing some robustness measure is sound. In this paper we address bi-objective versions of such location problems, in which the total cost, as well as the robustness associated with the demand, are optimized. A dominating set is constructed for these bi-objective nonlinear integer problems via the \(\varepsilon \)-constraint method. Computational results on test instances are presented, showing the feasibility of our approach to approximate the Pareto-optimal set.

A quadtree-based allocation method for a class of large discrete Euclidean location problems

A special data compression approach using a quadtree-based method is proposed for allocating very large demand points to their nearest facilities while eliminating aggregation error. This allocation procedure is shown to be extremely effective when solving very large facility location problems in the Euclidian space. Our method basically aggregates demand points where it eliminates aggregation-based allocation error, and disaggregates them if necessary. The method is assessed first on the allocation problems and then embedded into the search for solving a class of discrete facility location problems namely the p-median and the vertex p-center problems. We use randomly generated and TSP datasets for testing our method. The results of the experiments show that the quadtree-based approach is very effective in reducing the computing time for this class of location problems.

Metaheuristic applications on discrete facility location problems: a survey

This paper provides a detailed review of metaheuristic applications on discrete facility location problems. The objective of this paper is to provide a concise summary of solution approaches based on four commonly used metaheuristics: genetic algorithm, tabu search, particle swarm optimization and scatter search for different variants of the discrete facility location problem. Such a concise summary is expected to be useful for researchers interested in any of the major variants of discrete facility location problem as for each metaheuristic the paper provides a comprehensive review of different variants on which this metaheuristic has been applied, and the details of its implementation. Therefore, a research can exploit a method developed for another variant to solve the problem variant at hand. Based on our review of these papers, we also report some interesting observations, identify research gaps and highlight directions for future research.

Antennae Location Methodology for a Telecom Operator in India

This paper proposes a methodology for location of Base Stations of cellular radio networks in India with the objective of optimizing and automating the process of network planning. Operations of a large telecom operator in rural parts of India are studied. The Operator’s network planning team currently uses a manual approach to identify locations for the Base Stations. In this study the Base Station location problem is modeled as a discrete facility location problem and methodology that would help deal with conflicting network planning objectives i.e. cost, signal quality and coverage is presented and the best strategy under budget uncertainty is also explored. The model is run on data pertaining to one of the large States in India using CPLEX 10.0 [25] and OpenSolver [1]. Results indicate that, using the optimal approach, cost savings to the extent of 26% can be achieved. This is equivalent to a saving of INR 1 billion in the single State that is studied. Considering that the model can be replicated throughout the country, adoption of the proposed methodology can bring about substantial savings in the Operator’s Base Station infrastructure costs.

Solving a Location Problem of a Stackelberg Firm Competing with Cournot-Nash Firms

We study a discrete facility location problem on a network, where the locating firm acts as the leader and other competitors as the followers in a Stackelberg-Cournot-Nash game. To maximize expected profits the locating firm must solve a mixed-integer problem with equilibrium constraints. Finding an optimal solution is hard for large problems, and full-enumeration approaches have been proposed in the literature for similar problem instances. We present a heuristic solution procedure based on simulated annealing. Computational results are reported.

An Approximation Algorithm for the Facility Location Problem with Lexicographic Minimax Objective

We present a new approximation algorithm to the discrete facility location problem providing solutions that are close to the lexicographic minimax optimum. The lexicographic minimax optimum is a concept that allows to find equitable location of facilities serving a large number of customers. The algorithm is independent of general purpose solvers and instead uses algorithms originally designed to solve thep-median problem. By numerical experiments, we demonstrate that our algorithm allows increasing the size of solvable problems and provides high-quality solutions. The algorithm found an optimal solution for all tested instances where we could compare the results with the exact algorithm.