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Upgrading edges in the Maximal Covering Location Problem
Abstract
We study the upgrading version of the maximal covering location problem with edge length modifications on networks. This problem aims at locating p facilities on the vertices (of the network) so as to maximise coverage, considering that the length of the edges can be reduced at a cost, subject to a given budget. Hence, we have to decide on: the optimal location of p facilities and the optimal edge length reductions.
This problem is NP-hard on general graphs. To solve it, we propose three different mixed-integer formulations and a preprocessing phase for fixing variables and removing some of the constraints. Moreover, we strengthen the proposed formulations including valid inequalities. Finally, we compare the three formulations and their corresponding improvements by testing their performance over different datasets.
... The ECLP can be further subdivided into the Edge Set Location Covering Problem (ESCP) and the Edge Maximal Covering Location Problem (EMCLP) [10][11][12]. For the ESCP, its objective function is to minimize the construction costs or the number of facilities with the constraint that the entire demand must be totally covered [10,13]. In fact, this entire covering of edge demand on the network usually contains redundant covering on edge covering which means that some edge demand can be covered by multiple facilities at the same time. ...
... This redundant covering means an additional economic investment in the rescue spot and equipment, and that entire coverage is not economical. The EMCLP focuses on the maximization of the covering region with a fixed number of facilities, but cannot ensure the entire coverage of edge demand [13,14]. It is valuable that coverage rate on edge demand approximates to a preset range close to 100% with minimum redundancy. ...
... Baldomero-Naranjo et al. [11] studied the single-facility minmax regret maximal covering location problem with demand distributed along the edges. Baldomero-Naranjo et al. [13] further conducted another study on the upgrading version of the maximal covering location problem with edge length modifications on networks, by assuming that the length of the edges can be reduced at a cost. ...
The location of railway emergency rescue spots is facing diverse scenarios including the location of new facilities and optimization of existing layouts with limited or non-limited conditions. Generally there will be heavily redundant covering ability if all the edge demands on a network are fully covered. Here, we first proposed a near-full covering model to balance investment in the facility and the actual coverage rate, and successfully applied this model in the optimal location of railway emergency rescue spots under diverse scenarios. We also developed a feasible solution that can select an effective algorithm or a greedy algorithm based on the total consumed time. With the constraint of a fixed coverage rate threshold, a larger coverage radius may lead to fewer facilities and higher relative redundancy. Flexible designs of the important node set where all the elements must be selected and the exclusive node set where all the elements cannot be selected are carried out to construct several scenarios. The comparative analysis shows that the optimal solution is an obvious improvement on the existing emergency rescue spot layout in the real railway network. This study provides an alternative version of the edge covering problem, and shows a successful application in the location problem of railway rescue spots.
... In this case, changing the transportation mode (e.g. to a faster vehicle) may be a way to decrease the travel time thus upgrading the connection. The reader can refer to the recent paper by Baldomero-Naranjo et al. [2] for other examples of upgrading connections between demand nodes and centers in covering-type facility location problems. ...
... More recently, Baldomero-Naranjo et al. [2] investigated edge upgrading in the context of maximal covering facility location. Unlike the p-center problem, in which the coverage radius is endogenous, a maximum coverage radius is initially imposed and the goal is to install a certain number of facilities so that the maximum possible demand is covered. ...
... Other problems that have been considered in the upgrading context include the maximum shortest path interdiction problem Zhang et al. [27], hub location problem Blanco and Marín [4], and maximal covering problem Naranjo et al. [3]. ...
... In upgrading (downgrading) problems, the objective is to adjust network parameters within bounds to optimize the objective value of the modified network. Research in this area focuses on median locations Gassner (2008); Sepasian and Rahbarnia (2015); Plastria (2016), center location Sepasian (2018), hub location Blanco and Marín (2019), obnoxious median Afrashteh et al. (2020), and maximal covering Baldomero-Naranjo et al. (2022). ...
The single median facility location problem seeks to determine an optimal location to minimize the total weighted distance to existing demand points. However, in real-world scenarios, this optimal location can vary due to uncertainty of input data. In such situations, one can measure the robustness of the median location using stability radius, a concept in sensitivity analysis. In this paper, we first show that the stability radius of median location on tree admits a simple and tight lower bound. Furthermore, we investigate an upgrading model for this lower bound by adjusting vertex weights of the tree network within a specified budget. We also introduce a combinatorial approach capable of solving this problem efficiently.
... • Continuous upgrades over edges (arcs) considering that the length of the edges (arcs) can be modified by any increment subject to a prespecified budget. For covering problems, this has recently been considered by Baldomero-Naranjo et al. (2022), who proposed several mixed-integer programming formulations to solve the problem exactly. Other facility location problems where this approach has been studied include: the 1-center problem (Sepasian, 2018), the p-center problem with direct connection upgrades (Anton-Sanchez et al., 2023), and the p-median problem (Afrashteh et al., 2020, Espejo andMarín, 2023). ...
In this article, we study the complexity of the upgrading version of the maximal covering location problem with edge length modifications on networks. This problem is NP-hard on general networks. However, in some particular cases, we prove that this problem is solvable in polynomial time. The cases of star and path networks combined with different assumptions for the model parameters are analysed. In particular, we obtain that the problem on star networks is solvable in O(nlogn) time for uniform weights and NP-hard for non-uniform weights. On paths, the single facility problem is solvable in O(n^3) time, while the p-facility problem is NP-hard even with uniform costs and upper bounds (maximal upgrading per edge), as well as, integer parameter values. Furthermore, a pseudo-polynomial algorithm is developed for the single facility problem on trees with integer parameters.
... The underlying mathematical structure of the BRT investment problem also shows similarities to the more general network improvement problem. This problem consists of choosing edges (and nodes) in a network to be upgraded while minimizing costs or satisfying budget constraints (Krumke et al., 1998;Zhang et al., 2004;Baldomero-Naranjo et al., 2022). The problem has seen applications, e.g., in the area of road network optimization, where restricted resources can be used to upgrade edges in order to minimize the travel time between certain source-destination pairs (Lin and Mouratidis, 2015) or where roads can be upgraded to all-weather roads to improve the accessibility of health services (Murawski and Church, 2009). ...
... The underlying mathematical structure of the BRT investment problem also shows similarities to the more general network improvement problem. This problem consists of choosing edges (and nodes) in a network to be upgraded while minimizing costs or satisfying budget constraints [Krumke et al., 1998, Zhang et al., 2004, Baldomero-Naranjo et al., 2022. The problem has seen applications, e.g., in the area of road network optimization, where restricted resources can be used to upgrade edges in order to minimize the travel time between certain source-destination pairs [Lin and Mouratidis, 2015] or where roads can be upgraded to all-weather roads to improve the accessibility of health services [Murawski and Church, 2009]. ...
Bus Rapid Transit (BRT) systems can provide a fast and reliable service to passengers at low investment costs compared to tram, metro and train systems. Therefore, they can be of great value to attract more passengers to use public transport. This paper thus focuses on the BRT investment problem: Which segments of a single bus line should be upgraded such that the number of newly attracted passengers is maximized? Motivated by the construction of a new BRT line around Copenhagen, we consider a setting in which multiple parties are responsible for different segments of the line. As each party has a limited willingness to invest, we solve a bi-objective problem to quantify the trade-off between the number of attracted passengers and the investment budget. We model different problem variations: First, we consider two potential passenger responses to upgrades on the line. Second, to prevent scattered upgrades along the line, we consider different restrictions on the number of upgraded connected components on the line. We propose an epsilon-constraint-based algorithm to enumerate the complete set of non-dominated points and investigate the complexity of this problem. Moreover, we perform extensive numerical experiments on artificial instances and a case study based on the BRT line around Copenhagen. Our results show that we can generate the full Pareto front for real-life instances and that the resulting trade-off between investment budget and attracted passengers depends both on the origin-destination demand and on the passenger response to upgrades. Moreover, we illustrate how the generated Pareto plots can assist decision makers in selecting from a set of geographical route alternatives in our case study.
... Furthermore, Byrne & Kalcsics (2022) studied the conditional facility location where n facilities are located in a convex planar space (that also represents the spatial demand), and a single facility is about to be located considering all facilities and a convex polygon barrier which removes the demand. (For more applications of spatial optimization see: Baldomero-Naranjo et al., 2022 andLi et al., 2022.) Reviewing the mentioned papers above indicates that the literature does not address the underlying problem herein, whether in cloud seeding optimization or in modeling approach. . ...
... Although this paper belongs to the routing group, it is interesting to note that upgrading problems are also currently present in the location research community. As an example we can mention some recent upgrading location papers: Sepasian (2018) for the 1-center problem, Afrashteh et al. (2020) for the obnoxious -median location problem, Baldomero-Naranjo et al. (2022) for the maximal covering location problem, Blanco and Marín (2019) for the hub location problem and Espejo and Marín (2020) for the network -median problem. ...
In this article, we study the complexity of the upgrading version of the maximal covering location problem with edge length modifications on networks. This problem is NP‐hard on general networks. However, in some particular cases, we prove that this problem is solvable in polynomial time. The cases of star and path networks combined with different assumptions for the model parameters are analysed. In particular, we obtain that the problem on star networks is solvable in time for uniform weights and NP‐hard for non‐uniform weights. On paths, the single facility problem is solvable in time, while the ‐facility problem is NP‐hard even with uniform costs and upper bounds (maximal upgrading per edge), as well as, integer parameter values. Furthermore, a pseudo‐polynomial algorithm is developed for the single facility problem on trees with integer parameters.
Inspired by the core idea of the lattice Boltzmann method (LBM), which is successfully used in complex and nonlinear processes, we developed a lattice Boltzmann method-like (LBM-like) algorithm to effectively solve the maximal covering location problem with continuous- and inhomogeneous-edge demand on the complex network. The LBM-like algorithm developed has three key components, including the basic map, transfer function and effect function. The basic map is responsible for reasonably mapping complex networks with multiple branches and circles. Transfer functions are used to describe the complex covering process of the facility on the network, by splitting the entire covering process into several single-step covering processes, while the effect function is responsible for recording and processing the coverage effect of each covering process, based upon the requirement of an objective function. This LBM-like algorithm has good applicability to a complex network, intuitiveness, relatively low computational complexity, and open developability. Furthermore, the idea of the greedy algorithm was coupled with the LBM-like algorithm, to form two types of hybrid algorithms for improving the computational efficiency for the location problem, with multiple facilities, on a large-scale network. Finally, we successfully applied the LBM-like algorithm to the location problem of an emergency rescue spot on a real railway network, to underline the practicality of the proposed algorithm.
In this paper we present a generalization of the Graphical Travelling Salesman Problem (GTSP). Given a communication graph in which not all direct connections are necessarily possible, the Graphical Travelling Salesman Problem consists of finding the shortest tour that visits each node at least once. In this work, we assume the availability of a budget that allows to upgrade, i.e. reduce their traversal cost, some of the current connections and we propose the problem of designing the minimum cost tour using this budget. We propose and study a formulation for the problem, verifying that the polyhedron associated with the set of feasible solutions is a full-dimensional polytope. We present families of valid inequalities that reinforce the model and pre-processing techniques to reduce the number of variables of the formulation. To solve the problem, we propose a branch-and-cut algorithm that uses the introduced valid inequalities, as well as a heuristic to obtain good upper bounds and a tailor-made branching strategy. Comprehensive computational experiments on a new set of benchmark instances are presented to assess the performance of this exact method.
In the Maximum Capture Facility Location (MCFL) problem with a binary choice rule, a company intends to locate a series of facilities to maximize the captured demand, and customers patronize the facility that maximizes their utility. In this work, we generalize the MCFL problem assuming that the facilities of the decision maker act cooperatively to increase the customers' utility over the company. We propose a utility maximization rule between the captured utility of the decision maker and the opt-out utility of a competitor already installed in the market. Furthermore, we model the captured utility by means of an Ordered Median function (OMf) of the partial utilities of newly open facilities. We name this problem "the Cooperative Maximum Capture Facility Location problem" (CMCFL). The OMf serves as a means to compute the utility of each customer towards the company as an aggregation of ordered partial utilities, and constitutes a unifying framework for CMCFL models. We introduce a multiperiod non-linear bilevel formulation for the CMCFL with an embedded assignment problem characterizing the captured utilities. For this model, two exact resolution approaches are presented: a MILP reformulation with valid inequalities and an effective approach based on Benders' decomposition. Extensive computational experiments are provided to test our results with randomly generated data and an application to the location of charging stations for electric vehicles in the city of Trois-Rivi\`eres, Qu\`ebec, is addressed.
Network interdiction problems by deleting critical edges have wide applicatio ns. However, in some practical applications, the goal of deleting edges is difficult to achieve. We consider the maximum shortest path interdiction problem by upgrading edges on trees (MSPIT) under unit/weighted l1 norm. We aim to maximize the the length of the shortest path from the root to all the leaves by increasing the weights of some edges such that the upgrade cost under unit/weighted l1 norm is upper-bounded by a given value. We construct their mathematical models and prove some properties. We propose a revised algorithm for the problem (MSPIT) under unit l1 norm with time complexity O(n), where n is the number of vertices in the tree. We put forward a primal dual algorithm in O(n2) time to solve the problem (MSPIT) under weighted l1 norm, in which a minimum cost cut is found in each iteration. We also solve the problem to minimize the cost to upgrade edges such that the length of the shortest path is lower bounded by a value and present an O(n2) algorithm. Finally, we perform some numerical experiments to compare the results obtained by these algorithms.
This paper is concerned with the upgrading selective obnoxious p-median location problem on tree networks in which the existing customer points and the candidate facility locations are assumed to be two selective subsets of the vertex set of the underlying tree. The task is to augment the edge lengths within associated bounds and a budget constraint on the overall modification cost so that the optimal selective obnoxious p-median objective value is maximized under the new edge lengths. Exact optimal algorithms with polynomial time complexities are developed for cases p=1 and . Moreover, it is shown that if the bound constraints are dropped, then the problems under investigation can be solved in lower times.
In this paper we consider the covering problem on a network G = (V, E) with edge demands. The task is to cover a subset J ⊆ E of the edges with a minimum number of facilities within a predefined coverage radius. We focus on both the nodal and the absolute version of this problem. In the latter, facilities may be placed everywhere in the network. While there already exist polynomial time algorithms to solve the problem on trees, we establish a finite dominating set (i.e., a finite subset of points provably containing an optimal solution) for the absolute version in general graphs. Complexity and approximability results are given and a greedy strategy is proved to be a (1 + ln(|J|))‐approximate algorithm. Finally, the different approaches are compared in a computational study.
Covering problems constitute a fundamental family of facility location problems. This paper introduces a new exact algorithm for two important members of this family: (i) the maximal covering location problem (MCLP), which requires finding a subset of facilities that maximizes the amount of customer demand covered while respecting a budget constraint on the cost of the facilities; and (ii) the partial set covering location problem (PSCLP), which minimizes the cost of the open facilities while forcing a certain amount of customer demand to be covered. We study an effective decomposition approach to the two problems based on the branch-and-Benders-cut reformulation. Our new approach is designed for the realistic case in which the number of customers is much larger than the number of potential facility locations. We report the results of a series of computational experiments demonstrating that, thanks to this decomposition techniques, optimal solutions can be found very quickly for some benchmark instances with one hundred potential facility locations and involving up to 15 and 40 million customer demand points for the MCLP and the PSCLP, respectively.
In this paper, we introduce the Tree of Hubs Location Problem with Upgrading, a mixture of the Tree of Hubs Location Problem, presented by Contreras et. al (2010), and the Minimum Cost Spanning Tree Problem with Upgraded nodes, studied for the first time by Krumke (1999). In addition to locate the hubs, to determine the tree that connects the hubs and to allocate non-hub nodes to hubs, a decision has to be made about which of the hubs will be upgraded, taking into account that the total number of upgraded nodes is given. We present two different Mixed Integer Linear Programming formulations for the problem, tighten the formulations and generate several families of valid inequalities for them. A computational study is presented showing the improvements attained with the strengthening of the formulations and comparing them.
When deciding where to locate facilities (e.g., emergency points where an ambulance will wait for a call) that provide a service, it happens quite often that a customer (e.g., a person) can receive this service only if he/she is under a certain distance to the closest facility (e.g., the ambulance can arrive in less than 7 min at this person’s home). The problems that share this property receive the name of covering problems and have many applications (analysis of markets, archaeology, crew scheduling, emergency services, metallurgy, nature reserve selection, etc.). This chapter surveys the Set Covering Problem, the Maximal Covering Location Problem, and related problems and introduces a general model that has as particular cases the main covering location models. The main theoretical results in this topic as well as exact and heuristic algorithms are reviewed. A Lagrangian approach to solve the general model is detailed and, although the emphasis is on discrete models, some information on continuous covering is provided at the end of the chapter.
This is a survey on covering type location problems in a continuous environment, often also called minmax or centre location problems.
Location problems exist in the real world and they mainly deal with finding optimal locations for facilities in a network, such as net servers, hospitals, and shopping centers. The inverse location problem is also often met in practice and has been intensively investigated in the literature. As a typical inverse location problem, the inverse 1-median problem on tree networks with variable real edge lengths is discussed in this paper, which is to modify the edge lengths at minimum total cost such that a given vertex becomes a 1-median of the tree network with respect to the new edge lengths. First, this problem is shown to be solvable in linear time with variable nonnegative edge lengths. For the case when negative edge lengths are allowable, the NP-hardness is proved under Hamming distance, and strongly polynomial time algorithms are presented under and norms, respectively.
The maximum covering location model has been used extensively in analyzing locations for public service facilities. The model is extended to account for the chance that when a demand arrives at the system it will not be covered since all facilities capable of covering the demand are engaged serving other demands. An integer programming formulation of the new problem is presented. Several properties of the formulation are proven. A heuristic solution algorithm is presented and computational results with the algorithm are discussed. Directions for future study are also discussed.
The inverse p-median problem with variable edge lengths on graphs is to modify the edge lengths at minimum total cost with respect to given
modification bounds such that a prespecified set of p vertices becomes a p-median with respect to the new edge lengths. The problem is shown to be strongly NP{\mathcal{NP}}-hard on general graphs and weakly NP{\mathcal{NP}}-hard on series-parallel graphs. Therefore, the special case on a tree is considered: It is shown that the inverse 2-median
problem with variable edge lengths on trees is solvable in polynomial time. For the special case of a star graph we suggest
a linear time algorithm.
KeywordsLocation problem–Inverse optimization–
p-median–Complexity analysis
This paper discusses the inverse center location problem restricted on a tree with different costs and bound constraints.
The authors first show that the problem can be formulated as a series of combinatorial linear programs, then an O(∣V∣2 log ∣V∣) time algorithm to solve the problem is presented. For the equal cost case, the authors further give an O(∣V∣) time algorithm.
In this paper we present a cutting plane algorithm for the Set Covering problem. Cutting planes are generated by running an “exact” separation algorithm over the subproblems defined by suitably small subsets of the formulation constraints. Computational results on difficult small-medium size instances are reported.
This paper introduces the capacitated gradual and cooperative minimal covering location problem with distance constraints (cGC-MCLPD). The cGC-MCLPD extends the location literature by implementing the concepts of gradual and cooperative coverage in the context of undesirable facility location problem with distance constraints. It also allows for variable coverage radii and capacity of facilities to assess the effect of facility size on the network performance. For the defined problem, we first develop a nonlinear mathematical model which seeks to determine the number, location and size of facilities such that the total population covered is minimized while the overall service requirement is met. Next, we propose three integer linear programming formulations that can be solved with off-the-shelf solvers. The first two are linear approximations that are based on a separable programming approach and a tangent line approximation method. The third is an exact reformulation which uses a special network mapping technique. Upon investigating the impact of linearization approximation error on the performance of the first two formulations, we carry out numerical experiments to compare formulations with respect to their solution time and quality. Solving them for a set of reasonably large problem instances, we found that approximations outperform the exact reformulation since they prove to achieve higher quality solutions at the expense of an acceptable level of objective function value error. Overall, the formulations developed for the cGC-MCLPD constitute a powerful portfolio of facility location selection techniques, enabling decision-makers to select the most appropriate balance of solution quality and computational speed.
This paper addresses a version of the single-facility Maximal Covering Location Problem on a network where the demand is: i) distributed along the edges and ii) uncertain with only a known interval estimation. To deal with this problem, we propose a minmax regret model where the service facility can be located anywhere along the network. This problem is called Minmax Regret Maximal Covering Location Problem with demand distributed along the edges. Furthermore, we present two polynomial algorithms for finding the location that minimises the maximal regret assuming that the demand realisation is an unknown constant or linear function on each edge. We also include two illustrative examples as well as a computational study for the unknown constant demand case to illustrate the potential and limits of the proposed methodology.
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We study the problem of assigning doctors to existing, non-operational Primary Health Centers (PHCs). We do this in the presence of clear guidelines on the maximum population that can be served by any PHC, and uncertainties in the availability of the doctors over the planning horizon. We model the problem as a robust capacitated multi-period maximal covering location problem with server uncertainty. Such supply-side uncertainties have not been accounted for in the context of multi-period facility location in the extant literature. We present an MIP formulation of this problem, which turns out to be too difficult for an off-the-shelf solver like CPLEX. We, therefore, present several dominance rules to reduce the size of the model. We further propose a Benders decomposition based solution method with several refinements that exploit the underlying structure of the problem to solve it extremely efficiently. Our computational experiments show one of the variants of our Benders decomposition based method to be on average almost 1,000 times faster, compared to the CPLEX MIP solver, for problem instances containing 300 demand nodes and 10 facilities. Further, while the CPLEX MIP solver could not solve most of the instances beyond 300 demand nodes and 10 facilities even after 20 hours, two of our variants of Benders decomposition could solve instances upto the size of 500 demand nodes and 15 facilities in less than 0.5 hour, on average.
When deciding where to locate facilities (e.g., emergency points where an ambulance will wait for a call) that provide a service, it happens quite often that a customer (e.g., a person) can receive this service only if she is located less than a certain distance from the nearest facility (e.g., the ambulance can arrive in less than 7 min at this person’s home). The problems that share this property receive the name of covering problems and have many applications. (analysis of markets, archaeology, crew scheduling, emergency services, metallurgy, nature reserve selection, etc.). This chapter surveys the most relevant problems in this field: the Set Covering Problem, the Maximal Covering Location Problem, and related problems, In addition, it is introduced a general model that has as particular cases the main covering location models. The most important theoretical results in this topic as well as exact and heuristic algorithms are reviewed. A Lagrangian approach to solve the general model is detailed, and, although the emphasis is on discrete models, some information on continuous covering is provided at the end of the chapter.
A key aspect of the design of evolutionary and swarm intelligence algorithms is studying their performance. Statistical comparisons are also a crucial part which allows for reliable conclusions to be drawn. In the present paper we gather and examine the approaches taken from different perspectives to summarise the assumptions made by these statistical tests, the conclusions reached and the steps followed to perform them correctly. In this paper, we conduct a survey on the current trends of the proposals of statistical analyses for the comparison of algorithms of computational intelligence and include a description of the statistical background of these tests. We illustrate the use of the most common tests in the context of the Competition on single-objective real parameter optimisation of the IEEE Congress on Evolutionary Computation (CEC) 2017 and describe the main advantages and drawbacks of the use of each kind of test and put forward some recommendations concerning their use.
This paper studies the maximal covering location problem, assuming imprecise knowledge of all data involved. The considered problem is modeled from a fuzzy perspective producing suitable fuzzy Pareto solutions. Some properties of the fuzzy model are studied, which validate the equivalent mixed-binary linear multiobjective formulation proposed. A solution algorithm is developed, based on the augmented weighted Tchebycheff method, which produces solutions of guaranteed Pareto optimality. The effectiveness of the algorithm has been tested with a series of computational experiments, whose numerical results are presented and analyzed.
This paper discusses upgrading the 1-center problem on networks, which tries to change the lengths of the edges within certain bounds and find the best place for 1-center with respect to the new lengths so that the objective value is minimized. As this problem is NP-hard on general graphs, the problem is considered where the underlying graph is a tree. It is mentioned that this problem is solvable in polynomial time by solving a series of linear programs. A combinatorial algorithm with On2log(n) time complexity is proposed for the equal cost case, where n is the number of vertices of the tree. It is also shown that the problem is solvable in On2log(n)2 time on an unweighted tree, i.e., all vertex weights are equal to one, but the costs are arbitrary.
This paper introduces a very general discrete covering location model that accounts for uncertainty and time-dependent aspects. A MILP formulation is proposed for the problem. Afterwards, it is observed that most of the models existing in the literature related with covering location can be considered as particular cases of this formulation. In order to tackle large instances of this problem a Lagrangian relaxation based heuristic is developed. A computational study is addressed to check the potentials and limits of the formulation and some variants proposed for the problem, as well as to evaluate the heuristic. Finally, different measures to report the relevance of considering a multi-period stochastic setting are studied.
This paper addresses upgrading min-max spanning tree problem (MMST). Given a graph G(V,E), the aim of this problem is to modify edge weights under certain limits and given budget so that the MMST with respect to perturbed graph improves as much as possible. We present a complexity result for general non-decreasing cost functions. In special case, it is shown that the problem under linear and sum-type Hamming cost function can be solved in O(|E|2) and O(|E|log |E|log |V|) time, respectively.
In practical location problems on networks, the response time between any pair of vertices and the demands of vertices are usually indeterminate. This paper employs uncertainty theory to address the location problem of emergency service facilities under uncertainty. We first model the location set covering problem in an uncertain environment, which is called the uncertain location set covering model. Using the inverse uncertainty distribution, the uncertain location set covering model can be transformed into an equivalent deterministic location model. Based on this equivalence relation, the uncertain location set covering model can be solved. Second, the maximal covering location problem is investigated in an uncertain environment. This paper first studies the uncertainty distribution of the covered demand that is associated with the covering constraint confidence level α. In addition, we model the maximal covering location problem in an uncertain environment using different modelling ideas, namely, the (α, β)-maximal covering location model and the α-chance maximal covering location model. It is also proved that the (α, β)-maximal covering location model can be transformed into an equivalent deterministic location model, and then, it can be solved. We also point out that there exists an equivalence relation between the (α, β)-maximal covering location model and the α-chance maximal covering location model, which leads to a method for solving the α-chance maximal covering location model. Finally, the ideas of uncertain models are illustrated by a case study.
Problems aiming at finding budget constrained optimal upgrading schemes to improve network performance have received attention over the last two decades. In their general setting, these problems consist of designing a network and, simultaneously, allocating (limited) upgrading resources in order to enhance the performance of the designed network.
In this paper we address two particular optimal upgrading network design problems; in both cases, the sought-after layout corresponds to a spanning tree of the input network and upgrading decisions are to be taken on nodes. We design Mixed Integer Programming-based algorithmic schemes to solve the considered problems: Lagrangian relaxation approaches and branch-and-cut algorithms. Along with the designed algorithms, different enhancements, including valid inequalities, primal heuristic and variable fixing procedures, are proposed.
Using two set of instances, we experimentally compare the designed algorithms and explore the benefits of the devised enhancements. The results show that the proposed approaches are effective for solving to optimality most of the instances in the testbed, or manage to obtain solutions and bounds giving very small optimality gaps. Besides, the proposed enhancements turn out to be beneficial for improving the performance of the algorithms.
We study the planar maximum coverage location problem (MCLP) with rectilinear distance and rectangular demand zones in the case where "partial coverage" is allowed in its true sense, i.e., when covering part of a demand zone is allowed and the coverage accrued as a result of this is proportional to the demand of the covered part only. We pose the problem in a slightly more general form by allowing service zones to be rectangular instead of squares, thereby addressing applications in camera view-frame selection as well. More specifically, our problem, referred to as PMCLP-PCR (planar MCLP with partial coverage and rectangular demand and service zones), is to position a given number of rectangular service zones (SZs) on the two-dimensional plane to (partially) cover a set of existing (possibly overlapping) rectangular demand zones (DZs) such that the total covered demand is maximized. Previous studies on (planar) MCLP have assumed binary coverage, even when nonpoint objects such as lines or polygons have been used to represent demand. Under the binary coverage assumption, the problem can be readily formulated and solved as a binary linear program; whereas, partial coverage, although much more realistic, cannot be efficiently handled by binary linear programming, making PMCLP-PCR much more challenging to solve. In this paper, we first prove that PMCLP-PCR is NP-hard if the number of SZs is part of the input. We then present an improved algorithm for the single-SZ PMCLP-PCR, which is at least two times faster than the existing exact plateau vertex traversal algorithm. Next, we study multi-SZ PMCLP-PCR for the first time and prove theoretical properties that significantly reduce the search space for solving this problem, and we present a customized branch-and-bound exact algorithm to solve it. Our computational experiments show that this algorithm can solve relatively large instances of multi-SZ PMCLP-PCR in a short time. We also propose a fast polynomial time heuristic algorithm. Having optimal solutions from our exact algorithm, we benchmark the quality of solutions obtained from our heuristic algorithm. Our results show that for all the random instances solved to optimality by our exact algorithm, our heuristic algorithm finds a solution in a fraction of a second, where its objective value is at least 91% of the optimal objective in 90% of the instances (and at least 69% of the optimal objective in all the instances).
In this paper, the literature associated with the covering location problems addressing uncertainty under a fuzzy approach is reviewed. Specifically, the papers related to the most commonly applied models such as set covering location problem, maximal covering location problem, and hub covering location problem are examined. An annotated bibliography is presented in which such papers have been classified according to the following criteria: the fuzzy items considered in the proposed model, the type of problem addressed, the fuzzy approach applied, the method of resolution, and field of application considered. This research provides useful information that helps to identify some opportunities for the application of fuzzy approaches to the covering location problems.
We consider the 1-median problem with euclidean distances with uncertainty in the weights, expressed as possible changes within given bounds and a single budget constraint on the total cost of change. The upgrading (resp. downgrading) problem consists of minimizing (resp. maximizing) the optimal 1-median objective value over these weight changes. The upgrading problem is shown to belong to the family of continuous single facility location-allocation problems, whereas the downgrading problem reduces to a convex but highly non-differentiable optimization problem. Several structural properties of the optimal solution are proven for both problems, using novel planar partitions, the knapsack Voronoi diagrams, and lead to polynomial time solution algorithms.
Upgrading p-median problem is a problem of finding the best median of the given graph through modification of its parameters. The current paper develops a polynomial-time model to address this problem when the weights of vertices can be varying under a given budget. Moreover, in the case where the considering graph has a special structure, namely a path, a linear time algorithm will be proposed for solving the problem with uniform cost.
This paper addresses the problem of modifying the edge lengths of a tree in minimum total cost such that a prespecified vertex becomes the 1-center of the perturbed tree. This problem is called the inverse 1-center problem on trees. We focus on the problem under Chebyshev norm and Hamming distance. From special properties of the objective functions, we can develop combinatorial algorithms to solve the problem. Precisely, if there does not exist any vertex coinciding with the prespecified vertex during the modification of edge lengths, the problem under Chebyshev norm or bottleneck Hamming distance is solvable in O(nlogn) time, where n+1 is the number of vertices of the tree. Dropping this condition, the problem can be solved in O(n^2) time.
This paper considers two covering location problems on a network where the demand is distributed along the edges. The first is the classical maximal covering location problem. The second problem is the obnoxious version where the coverage should be minimized subject to some distance constraints between the facilities. It is first shown that the finite dominating set for covering problems with nodal demand does not carry over to the case of edge based demands. Then, a solution approach for the single facility problem is presented. Afterwards, the multi-facility problem is discussed and several discretization results for tree networks are presented for the case that the demand is constant on each edge; unfortunately, these results do not carry over to general networks as a counter example shows. To tackle practical problems, the conditional version of the problem is considered and a greedy heuristic is introduced. Afterwards, numerical tests are presented to underline the practicality of the algorithms proposed and to understand the conditions under which accurate modeling of edge-based demand and a continuous edge-based location space are particularly important.
This paper is concerned with a reverse obnoxious (undesirable) center location problem on networks in which the aim is to modify the edge lengths within an associated budget such that a predetermined facility location on the underlying network becomes as far as possible from the existing customer points under the new edge lengths. Exact combinatorial algorithms with linear time complexities are developed for the problem under the weighted rectilinear norm and the weighted Hamming distance. Furthermore, it is shown that the problem with integer decision variables can also be solved in linear time.
In this article, we consider the cooperative maximum covering location problem on a network. In this model, it is assumed that each facility emits a certain “signal” whose strength decays over distance according to some “signal strength function.” A demand point is covered if the total signal transmitted from all the facilities exceeds a predefined threshold. The problem is to locate facilities so as to maximize the total demand covered. For the 2-facility problem, we present efficient polynomial algorithms for the cases of linear and piecewise linear signal strength functions. For the p-facility problem, we develop a finite dominant set, a mixed-integer programming formulation that can be used for small instances, and two heuristics that can be used for large instances. The heuristics use the exact algorithm for the 2-facility case. We report results of computational experiments. © 2014 Wiley Periodicals, Inc. NETWORKS, 2014
The location set-covering problem (LSCP) and the maximal covering location problem (MCLP) have been the subject of considerable interest. As originally defined, both problems allowed facility placement only at nodes. This paper deals with both problems for the case when facility placement is allowed anywhere on the network. Two theorems are presented that show that when facility placement is unrestricted, for either the LSCP or MCLP at least one optimal solution exists that is composed entirely of points belonging to a finite set of points called the network intersect point set (NIPS). Optimal solution approaches to the unrestricted site LSCP and MCLP problems that utilize the NIPS and previously developed solution methodologies are presented. Example solutions show that considerable improvement in the amount of coverage or the number of facilities needed to insure total coverage can be achieved by allowing facility placement along arcs of the network. In addition, extensions to the arc-covering model and the ambulance-hospital model of ReVelle, Toregas, and Falkson are developed and solved.
Graphs with weights and delays associated with their edges and/or vertices are often used to model communication and signal flow networks. Network performance can be improved by upgrading the network vertices. Such an upgrade reduces the edge/vertex delays and comes at a cost. We study different formulations of this network performance improvement problem and show that these are NP-hard. We then consider one of the formulations and develop polynomial time algorithms for some special cases and pseudopolynomial time algorithms for others.
While classical location problems deal with finding optimal locations for facilities, the task of the corresponding upgrading (down-grading) version is to change the underlying network within certain bounds such that the optimal objective value that can be obtained in the modified network is as good (bad) as possible. In this paper we al-low to change the vertex weights within given bounds such that a linear budget constraint is satisfied. For the upgrading 1-median problem an O(n 2) time algorithm is suggested. The downgrading 1-median problem is shown to be solvable in polynomial time. For the special case of a tree a concavity property leads to an O(n log n) time algorithm.
Network location models have been used extensively for siting public and private facilities. In this paper, we investigate a model that simultaneously optimizes facility locations and the design of the underlying transportation network. Motivated by the simple observation that changing the network topology is often more cost-effective than adding facilities to improve service levels, the model has a number of applications in regional planning, distribution, energy management, and other areas. The model generalizes the classical simple plant location problem. We show how the model can be solved effectively. We then use the model to analyze two potential transportation planning scenarios. The fundamental question of resource allocation between facilities and links is investigated, and a detailed sensitivity analysis provides insight into the model's usefulness for aiding budgeting and planning decisions. We conclude by identifying promising research directions.
This paper presents the flow cost lowering problem (FCLP), which is an extension to the integral version of the well-known minimum cost flow problem (MCFP). While in the MCFP the flow costs are fixed, the FCLP admits lowering the flow cost on each arc by upgrading the arc. Given a flow value and a bound on the total budget which can be used for upgrading the arcs, the goal is to find an upgrade strategy and a flow of minimum cost. The FCLP is shown to be NP-hard even on series–parallel graphs. On the other hand the paper provides a polynomial time approximation algorithm on series–parallel graphs.
We consider a generalization of the maximal cover location problem which allows for partial coverage of customers, with the degree of coverage being a non-increasing step function of the distance to the nearest facility. Potential application areas for this generalized model to locating retail facilities are discussed.We show that, in general, our problem is equivalent to the uncapacitated facility location problem. We develop several integer programming formulations that capitalize on the special structure of our problem. Extensive computational analysis of the solvability of our model under a variety of conditions is presented.
In this paper we consider a reverse center location problem in which we wish to spend as less cost as possible to ensure that
the di- stances from a given vertex to all other vertices in a network are within given upper bounds. We first show that this
problem is NP-hard. We then formulate the problem as a mixed integer programming problem and propose a heuristic method to
solve this problem approximately on a spanning tree. A strongly polynomial method is proposed to solve the reverse center
location problem on this spanning tree.
We introduce the Upgrading Shortest Paths Problem, a new combinatorial problem for improving network connectivity with a wide
range of applications from multicast communication to wildlife habitat conservation. We define the problem in terms of a network
with node delays and a set of node upgrade actions, each associated with a cost and an upgraded (reduced) node delay. The
goal is to choose a set of upgrade actions to minimize the shortest delay paths between demand pairs of terminals in the network,
subject to a budget constraint. We show that this problem is NP-hard. We describe and test two greedy algorithms against an
exact algorithm on synthetic data and on a real-world instance from wildlife habitat conservation. While the greedy algorithms
can do arbitrarily poorly in the worst case, they perform fairly well in practice. For most of the instances, taking the better
of the two greedy solutions accomplishes within 5% of optimal on our benchmarks.
This paper concerns the reverse 2-median problem on trees and the reverse 1-median problem on graphs that contain exactly one cycle. It is shown that both models under investigation can be transformed to an equivalent reverse 2-median problem on a path. For this new problem an O(nlogn) algorithm is proposed, where n is the number of vertices of the path. It is also shown that there exists an integral solution if the input data are integral.
The metaheuristic heuristic concentration (HC) is applied here to the solution of large instances of the maximal covering location problem with high percentage coverage. In these instances, exact methods may be too cumbersome for practical use, and heuristics can allow faster solution times with near-optimal results. We examined the performance of HC based on its ability to approach the optimal solutions to these problems and the run times of the algorithm compared to LP-IP runtimes. Exact solutions, generated by linear programming and branch and bound, provided a benchmark for comparison when the LP-IP problems could be run to completion. In all cases, HC found solutions with objective values no worse than 0.543% below the best known LP-IP objective value. In several instances, LP-IP runtime ballooned to as much as 38.5h, while HC took no longer than 1.6h in any instance. In one particular instance, LP-IP took 38.5h to terminate, while HC found a near-optimal solution (within 0.306% of optimality) in only 25min. Furthermore, in 62.5% of the runs, the second stage of HC improved on the first stage 1-opt algorithm.
Location problems exist extensively in the real world and they mainly deal with finding optimal locations for facilities. However, the reverse location problem is also often met in practice, in which the facilities may already exist in a network and cannot be moved to a new place, the task is to improve the network within a given budget such that the improved network works as efficient as possible. This paper is dedicated to the problem of how to use a limited budget to modify the lengths of the edges on a cycle such that the overall sum of the weighted distances of the vertices to the respective closest facility of two prespecified vertices becomes as small as possible (shortly, R2MC problem). It has already been shown that the reverse 2-median problem with edge length modification on general graphs is strongly NP-hard. In this paper, we transform the R2MC problem to a reverse 3-median problem on a path and show that this problem can be solved efficiently by strongly polynomial algorithm.
This paper is concerned with bottleneck weight reduction problems (WRPs) stated as follows. We are given a finite set E, a class F of nonempty subsets of E, a weight w:E→R+ and a cost c:E→R+. For each e∈E, c(e) stands for the cost of reducing weight w(e) by one unit. For each subset F∈F, the bottleneck weight of F is w(F)=mine∈Fw(e). The weight of the family F is the maximum of w(F) for all F in F. The problem is to determine new weights x(e)⩽w(e) such that the weight of F is minimized under the constraint that the overall reduction cost does not exceed a given budget B. Similarly to capacity expansion problems, WRPs include NP-hard problems. A WRP can be formulated as a parametric optimization problem over all transversal sets T of the class F. This leads to (strongly) polynomial solution procedures for special systems F. In particular we outline a polynomial algorithm in the case when F is the class of all spanning trees in an undirected graph.
The gradual covering location problem seeks to establish facilities on a network so as to maximize the total demand covered, allowing partial coverage. We focus on the gradual covering location problem when the demand weights associated with nodes of the network are random variables whose probability distributions are unknown. Using only information on the range of these random variables, this study is aimed at finding the “minmax regret” location that minimizes the worst-case coverage loss. We show that under some conditions, the problem is equivalent to known location problems (e.g. the minmax regret median problem). Polynomial time algorithms are developed for the problem on a general network with linear coverage decay functions.
The objective of this paper is to identify the most promising sets of closest assignment constraints in the
literature of Discrete Location Theory, helping the authors in the field to model their problems when clients
must be assigned to the closest plant inside an Integer Programming formulation. In particular, constraints leading to weak Linear Programming relaxations should be avoided if no other good property
supports their use. We also propose a new set of constraints with good theoretical properties.
This article deals with the reverse 1-median problem on graphs with positive vertex weights. The problem is proved to be strongly NP-hard even in the case of bipartite graphs and not approximable within a constant factor (unless P = NP). Furthermore, a linear time algorithm for the reverse 1-median problem on a cycle with linear cost functions (RMC) is developed. It is also shown that there exists an integral optimal solution of RMC if the input data are integral. © 2006 Wiley Periodicals, Inc. NETWORKS, Vol. 48(1), 16-23 2006
The inverse p-median problem consists in changing the weights of the customers of a p-median location problem at minimum cost such that a set of p prespecified suppliers becomes the p-median. The cost is proportional to the increase or decrease of the corresponding weight. We show that the discrete version of an inverse p-median problem can be formulated as a linear program. Therefore, it is polynomially solvable for fixed p even in the case of mixed positive and negative customer weights. In the case of trees with nonnegative vertex weights, the inverse 1-median problem is solvable in a greedy-like fashion. In the plane, the inverse 1-median problem can be solved in O(nlogn) time, provided the distances are measured in l1- or l∞-norm, but this is not any more true in R3 endowed with the Manhattan metric.
The convex ordered median problem is a generalization of the median, the k-centrum or the center problem. The task of the associated inverse problem is to change edge lengths at minimum cost such that a given vertex becomes an optimal solution of the location problem, i.e., an ordered median. It is shown that the problem is NP-hard even if the underlying network is a tree and the ordered median problem is convex and either the vertex weights are all equal to 1 or the underlying problem is the k-centrum problem. For the special case of the inverse unit weight k-centrum problem a polynomial time algorithm is developed.