P-median Problem Research Articles

Less is more: adjusting convergence of Cooper’s algorithm in the continuous location-allocation problem

This paper proposes a variant of the well-known heuristic by Cooper for continuous location-allocation problems in general, and in particular, the continuous p-median problem. This algorithm alternates between a location phase where the allocations are fixed and an allocation phase where the locations of the p new facilities are fixed. The main idea of the proposed variant is to partially solve the p independent one-median problems in the location phase (instead of solving optimally) in order to induce a larger number of cycles of location-allocation. In effect, we are trying to alter the descent path of the algorithm by slowing the convergence rate using smaller descent steps in the location phase. Computational experiments show that this approach is highly effective in finding alternate descent paths that lead to “deeper” local minima. The best improvements are obtained with random starting locations of the p facilities, and the larger values of p that were tested.

DETERMINING THE OPTIMAL TEMPORARY WASTE DISPOSAL SITES IN THE ALANG-ALANG LEBAR SUB-DISTRICT PALEMBANG USING THE P-CENTRE LOCATION PROBLEM AND P-MEDIAN PROBLEM MODELS

The rapid development of Palembang City comes with an increase in population and a proportionate increase in waste. Providing Temporary Waste Disposal Sites (TWDS) with ideal locations is one way to address the waste problem in Palembang City. The location of the existing TWDS could be more regular and optimal. The problem of determining the optimal TWDS location can be solved by optimization science, as classified in the Set Covering Problem (SCP) model. The SCP model is divided into the -Center Location Problem and -Median Problem models. This study aims to determine the optimal locations for TWDS in the Alang-Alang Lebar Sub-District, Palembang City, by comparing the results of the p-Center Location Problem and p-Median Problem models. Initially, the Alang-Alang Lebar Sub-District had 33 TWDS. After formulating the Set Covering Location Problem and Maximal Covering Location Problem models, we obtain the optimal solution, which we then solve using the -Center Location Problem and -Median Problem models. Based on the results and discussion, the optimal TWDS can meet the demand of each village in the Alang-Alang Lebar Sub-District. The -Center Location Problem and -Median Problem models produce the same optimal TWDS, namely TWDS Pramuka 2 Street and around, TWDS Colonel Sulaiman Amin Street, TWDS Talang Kelapa Ujung, and TWDS Beside Soekarno Hatta Street. This study recommends using both models to determine the optimal TWDS.

Exact methods and a variable neighborhood search for the robust capacitated [formula omitted]-median problem

The capacitated p-median problem (CPMP) involves placing p identical facilities in a network and assigning customer nodes to these facilities to satisfy all customer demands with minimal transportation costs. In practical applications, demand and distance parameters are often uncertain during the planning process, leading to infeasible or excessively costly solutions if these uncertainties are disregarded. This paper addresses the robust CPMP (RCPMP), which incorporates demand uncertainty into the problem using the robust optimization paradigm. We propose a general framework to model and solve the RCPMP, considering different polyhedral uncertainty sets, namely the cardinality-constrained and the knapsack sets. We develop exact approaches encompassing compact models, different families of valid inequalities, and branch-and-cut and branch-and-price algorithms, exploring both implemented uncertainty sets and problem structure. Furthermore, we implement an efficient Variable Neighborhood Search (VNS) heuristic to solve these robust variants, which incorporates state-of-the-art algorithms, parallelization techniques, and optimized data structures. Computational experiments using adapted benchmark instances with up to 400 nodes indicate the effectiveness of the proposed approaches. The results show that using parallelization and hash tables within the VNS heuristic promotes significant performance improvements and yields near-optimal solutions for most instances, as well as outperforming the exact approaches in several instances where the optimal solution was not found. Moreover, these results highlight the benefits of using robust solutions in practical settings, especially when considering different uncertainty sets to generate solutions with advantageous trade-offs between cost and risk.

Integrated optimization of vessel dispatching and empty container repositioning considering turnover time uncertainty

The global trade disproportion results in the accumulation of containers in import-dominated ports and shortages in export-dominated ports, causing congestion and high freight costs, thus hindering maritime shipping economy development. To address these issues, this study develops a stochastic programming model considering uncertain container turnover times. The model integrates decisions for vessel deployment and empty container repositioning over multiple planning periods through a two-stage decision process, aiming to minimize the total cost, including vessel deployment, container leasing, and penalty costs for unfulfilled demand. By formulating the scenario selection problem as a p-median problem, we effectively reduce the model size. We propose an accelerated Benders decomposition algorithm which leverages the independence of sub-problems in the second stage to enable parallel computation. Numerical experiments show that our Benders decomposition algorithm improves solution speed by over 63% compared to the Gurobi optimization solver. Furthermore, our integrated optimization approach proves to be more cost-effective than the reactive method used by shipping lines, achieving an average cost savings of 0.72%. Additionally, our method of constructing turnover time scenarios to address uncertainty saves approximately 0.45% in costs compared to using the probability distribution of container turnover time.

Optimizing Defibrillator Deployment with Bus-Mounted Automated External Defibrillator

Objectives Early defibrillation with an automated external defibrillator (AED) can effectively improve the survival rate of patients with out-of-hospital cardiac arrest (OHCA). Placing AEDs in public locations can reduce the defibrillation response interval from collapse to defibrillation. Most public AEDs are currently placed in a stationary way (S-AED) with limited coverage area. Bus mounted AED (B-AED) can be delivered directly to the demand point. Although B-AEDs are only available during bus operating hours, they provide greater coverage area. When the number of available AEDs is insufficient, better coverage may be achieved by placing a portion of AEDs as B-AEDs. Our purpose is developing a model to determine the optimal locations of B-AEDs and S-AEDs with a predetermined number of available AEDs. The goal is to maximize the total coverage level of all demand points. Methods We proposed a joint location model to place B-AEDs and S-AEDs based on the p-median problem (JPMP). Using data from Chang’an District, Xi’an City, China, we determined the optimal AED deployment. The performance of JPMP was compared with several other models. The coverage results of JPMP are analyzed in details, including the quantity assignment, coverage level, and geographical location of B-AEDs and S-AEDs. The impact of the bus departure intervals on coverage was also discussed. Results The use of B-AEDs results in an average 98.43% increase in the number of covered demand points, and an average 74.05% increase in total coverage level. In optimal AED deployment, B-AEDs coverage follows an inverted U-shaped curve with increasing number of available AEDs. It begins to decrease when all demand points during the operating hours are covered. With a constant number of available AEDs, the total coverage level increases and then decreases as the bus departure interval increases. The larger the number of available AEDs, the smaller the optimal departure interval. Conclusions With a given number of available AEDs, combinational deployment of B-AEDs and S-AEDs significantly improves the coverage level. B-AEDs are recommended when AEDs are insufficient. If more AEDs are available, better coverage can be obtained with reasonable location of S-AEDs and B-AEDs.

Quadratic p-Median Problem: A Bender’s Decomposition and a Meta-Heuristic Local-Based Approach

In this paper, the quadratic p-median optimization problem is discussed, where the goal is to connect users to a selected group of facilities (emergency services, telecommunications servers, healthcare facilities) at the lowest possible cost. The problem is aimed at minimizing the cost of connecting these selected facilities. The costs are symmetric, meaning connecting two different points is the same in both directions. This problem extends the traditional p-median problem, a combinatorial problem used in various fields like facility location, network design, transportation, supply chain networks, emergency services, healthcare, and education planning. Surprisingly, the quadratic version has not been thoroughly considered in the literature. The paper highlights the formulation of two mixed-integer quadratic programming models to find optimal solutions to this problem. One model is a classic formulation, and the other is based on set cover theory. Linear versions and Bender’s decomposition formulations for each model are also derived. A Bender’s decomposition is solved using an algorithm that adds constraints during each iteration to improve the solution. Lazy constraints in the Gurobi solver’s branch and cut algorithm are dynamically added whenever a mixed-integer programming solution is found. Additionally, an efficient local search meta-heuristic is proposed that usually finds optimal solutions for tested instances. Challenging instances with up to 60 facilities and 2000 users are successfully solved. Our results show that Bender’s models with lazy constraints are the most effective for Euclidean and random test cases, achieving optimal solutions in less CPU time. The meta-heuristic also finds near-optimal solutions rapidly for these cases.

Spatial matching of emergency shelters to the distribution of residents in the light of transport behaviours of evacuees during war: the case of Suwałki

The aim of the article was the assessment of the spatial matching of existing shelters (supply) to the distribution of residents in Suwałki (demand), considering their declared transport behaviours while evacuating during war. The analysis was conducted based on the locations of existing emergency shelters using data on population distribution (registration data with building accuracy). Spatial alignment was determined using the P-Median problem and E2SFCA. In terms of establishing vehicular or pedestrian travel time, the Manhattan metric based on the urban road network model was utilised. A model of vehicle movement speed was then constructed, while a constant speed was assumed for pedestrian movement. Additionally, survey data on the transport behaviour of inhabitants of Suwałki in the case of war were conducted in 2023. The study concluded that the population residing within the city limits should evacuate on foot, and that prior training on the evacuation process is especially necessary for those who reside in less populated areas of the city. The analyses also showed that existing emergency shelters are overly dispersed, making management difficult for emergency services. Since the current capacity of emergency shelters is not sufficient for the number and distribution of inhabitants of Suwałki, the most practical significance of this article in this respect is to indicate to the authorities the optimal number and location for emergency shelters (to improve the evacuation process).

A holistic approach to introducing a light electric freight vehicle (LEFV) system in a historic urban environment: The case of Quito

In this paper we propose a comprehensive approach for introducing a first/last mile system based on Light Electric Freight Vehicles (LEFVs) in the historical centre of Quito (Ecuador). The approach includes four crucial steps for successful launching and establishment of a LEFVs system. The first step considers organizational aspects by proposing a number of alternative cooperative business models. In the second step, the LEFVs fleet mix problem is defined as a Multicriteria Decision Making Problem (MCDM) and solved by the Analytic Network Process (ANP). The third step includes the problem of optimal micro hub location as well as the problem of optimal routing of LEFVs. The micro hub location problem is modelled as a capacitated p-median problem and solved by an exact solution algorithm. The problem of optimal LEFVs routing is defined as the capacitated pickup and delivery with time windows addressed by a web-based optimization tool. The fourth step includes an assessment of the existing policy framework and proposing a set of additional legal instruments for facilitating the LEFVs market uptake. Based on the proposed four-step approach, the first/last mile system is currently being implemented in the historical centre of Quito. The initial results appear promising regarding the success of the new LEFVs first/last mile system.

The Classical p-median Problem as a Surrogate Model in Hub Location

The Classical p-median Problem as a Surrogate Model in Hub Location

Dispersed starting solutions in facility location: The case of the planar [formula omitted]-median problem

There are many planar multiple facilities location problems for which the optimal locations tend to be spread out. The most popular of these is the planar p-median problem. With this in mind, we propose several procedures to generate sparse configurations as starting solutions. The proposed procedures are easy to implement, and can be used as modules combined in different sequences within heuristics such as a recent trajectory-based procedure that we tested in this paper.The procedures are tested experimentally on a set of 24 large problem instances with up to 10,000 demand points and 100 facilities. We are able to demonstrate that the sparse starting solutions generated by the new procedures lead to significant improvements of final p-median solutions.

Electric vehicle charging stations: Model, algorithm, simulation, location, and capacity planning

The transition to sustainable transportation is imperative in mitigating environmental impacts, with electric vehicles (EVs) at the forefront of this shift. Despite their environmental benefits, the global adoption of EVs is curtailed by challenges such as nascent battery technology, high costs, and insufficient charging infrastructure. This study addresses the optimizing electric vehicle charging station (EVCS) locations as a critical step toward enhancing EV adoption rates. Thus, establishing efficient charging stations is critical to meet the increasing demand. By integrating location modeling with demand forecasts and market penetration, we propose a comprehensive approach to determine optimal locations and capacities for EVCS. Firstly, review existing literature, highlighting the significance of facility location models in optimizing EV charging infrastructure and identifying gaps in addressing demand and market penetration. Our methodology uses a genetic algorithm to solve the p-median problem for location selection and Arena 14 simulation software to model station traffic and optimize charging unit types and quantities. The model prioritizes public areas, considering potential demand points and station locations to propose optimal charging areas. Results indicate that our model minimizes travel distances and waiting times, offering a scalable solution adaptable to future EV market growth. This study contributes to the field by presenting a sustainable and economical model for EVCS placement and capacity planning, underlining the importance of a robust charging network in the broader adoption of electric transportation. The findings suggest that proactive infrastructure development, guided by accurate demand predictions and optimized location strategies, can significantly enhance the feasibility and attractiveness of EVs, supporting global efforts towards a cleaner, more sustainable transportation system.

On the Budgeted Priority p-Median Problem in High-Dimensional Euclidean Spaces

On the Budgeted Priority p-Median Problem in High-Dimensional Euclidean Spaces

A matheuristic for locating obnoxious facilities

Facilities such as waste plants or wind turbines are often referred to as obnoxious facilities because they negatively affect their nearby environment, for example, through noise or pollution. In the obnoxious p-median problem, a set of clients and a set of potential locations for obnoxious facilities are given. From the set of potential locations, p facilities must be opened. The goal is to place the facilities far away from the clients to avoid high negative effects. Existing approaches for this planning problem are either not scalable to large instances or not flexible in considering practical constraints that often arise in real-world settings. To address these limitations, we propose a matheuristic for the obnoxious p-median problem. First, the matheuristic generates diverse initial solutions, allowing an effective exploration of the solution space. Second, it iteratively improves these initial solutions using enhanced mathematical models. We additionally propose a clustering-based scaling technique to tackle large instances. Thus, our matheuristic is scalable to instances involving thousands of clients and potential locations and is flexible to incorporate practical constraints. Our computational results show that our matheuristic outperforms the leading metaheuristics from the literature on large instances and is competitive with the leading metaheuristics on small and medium instances. The features of our matheuristic can be generalized and applied to related planning problems.

The Hamiltonian p-median problem: Polyhedral results and branch-and-cut algorithms

In this paper we study the Hamiltonian p-median problem, in which we are given an edge-weighted graph and we are asked to determine p vertex-disjoint cycles spanning all vertices of the graph and having minimum total weight. We introduce two new families of valid inequalities for a formulation of the problem in the space of edge variables. Each one of the families forbids solutions to the 2-factor relaxation of the problem that have less than p cycles. The inequalities in one of the families are associated with large cycles of the underlying graph and generalize known inequalities associated with Hamiltonian cycles. The other family involves inequalities for the case with p=n/3, associated with edge cuts and multi-cuts whose shores have specific cardinalities. We identify inequalities from both families that define facets of the polytope associated with the problem. We design branch-and-cut algorithms based on these families of inequalities and on inequalities associated with 2-opt moves removing sub-optimal solutions. Computational experiments on benchmark instances show that the proposed algorithms exhibit a comparable performance with respect to existing exact methods from the literature. Moreover the algorithms solve to optimality new instances with up to 400 vertices.

A parallel variable neighborhood search for [formula omitted]-neighbor facility location problems

In this paper, we employ the less is more approach to develop a Parallel Variable Neighborhood Search (VNS) algorithm for the α-neighbor p-center problem (αNpCP) and the α-neighbor p-median problem (αNpMP). The αNpCP and the αNpMP are generalizations of the p-center (pCP) and p-median (pMP) problems, respectively. In the α-neighbor problems, one seeks to open p facilities and assign each of the n customers to their closest α ones. The objective is to minimize the maximum distance of a customer to its αth facility, in the case of the αNpCP, and the sum of the distances from each customer to their α nearest facilities, in the case of the αNpMP. Our VNS adapts simple but efficient algorithms and data structures from the pCP and pMP literature to the αNpCP and αNpMP context. We also introduce an updated objective function for the αNpCP, which adds more information to the solution cost and helps the VNS to escape from local optima. Several experimental tests show that our VNS outperforms more complex state-of-the-art algorithms. Regarding the αNpCP, on 120 instances derived from the OR-library set, our algorithm improved best-known solutions for 22, with an average improvement of 34.26%; the overall gap on the 120 instances is 6.18% in favor of our algorithm. Moreover, on 231 instances derived from the TSPLIB set, we improved the solutions for 115, with an average improvement of 5.30%, and an overall improvement gap of 2.47% for all 231 instances. Considering the αNpMP results, our heuristic obtained better results than a heuristic from literature in all 80 instances tested, finding optimal solutions in all these instances.

Multi-objective Colliding Bodies Optimization Algorithm for the Obnoxious p-median Problems

The obnoxious p-median problem consists of locating p facilities among a set of sites such that the sum distance from any demand to its nearest facility and the dispersion among facilities are maximized. In this paper, the multi-objective colliding bodies optimization algorithm (MOCBO) is utilized to obtain the trade-off curve of the obnoxious p-median problems. The performance of the developed optimization method is investigated for locating obnoxious facilities through two case studies to maximize the two conflicting objectives. The performance of the MOCBO algorithm is further compared with those of the MPSO and NSGA-II algorithms representative of the state of the art in the field of multi-objective optimization. In this study, the MOCBO algorithm showed suitable convergence performance and generalization abilities compared to the MPSO and NSGA-II algorithms.

Police service district planning

We propose a new framework to address the territory design problem of emergency services in collaboration with two police authorities in Europe. Our framework serves as a strategic decision support system to assess different districting layouts, department locations, staffing decisions and dispatching strategies. First, we introduce a novel modification of the p-median problem with a combined approach to the contiguity and compactness of district layouts solvable by a commercial solver. Second, we utilize a new discrete event simulation that accounts for the variability of spatial and temporal incident patterns and driving times to evaluate the district layouts according to several criteria based upon up to 1.8 million historical incidents. Our simulation results demonstrate that our proposed district layouts can lead to a reduction of the response time by up to 14.52% while also lowering the dispatch time, the overall driving time, and the number of unanswered calls for service. Additionally, we examine the computational complexity of optimally locating district centers and analyze the more restricted problem of optimally reassigning districts to fixed district centers.

Impacts of spatial imputation on location-allocation problem solutions

Georeferenced data often contain missing values, and such missing values can considerably affect spatial modeling. A spatial location model can also suffer from this issue when there are missing values in its geographic distribution of weights. Although general imputation approaches have been developed, one distinguishing fact here is that spatial imputation generally performs better for georeferenced data because it can reflect a fundamental property of those data, that is, spatial autocorrelation or spatial dependency. This paper explores how spatial imputation exploiting spatial autocorrelation can contribute to estimating missing values in a weights surface for location modeling and subsequently improve solutions for spatial optimization, specifically p-median problems using a spatially imputed weights surface. This paper examines two spatial imputation methods, ordinary co-kriging and Moran eigenvector spatial filtering. Their results are compared with conventional linear regression, essentially Expectation-Maximization algorithm results for independent observations of Gaussian random variable cases. Simulation experiments show that spatial imputation produces better results for georeferenced data than simply ignoring any missing values and non-spatial imputation, and appropriately imputed values can enhance spatial optimization solutions, regardless of the number of medians, p.

Optimizing health service location in a highly urbanized city: Multi criteria decision making and P-Median problem models for public hospitals in Jeddah City, KSA.

Rapid urbanization and population growth have increased the need for optimizing the location of health services in highly urbanized countries like Kingdom of Saudi Arabia (KSA). This study employs a multiple-criteria decision making (MCDM) approach, e.g., fuzzy overlay technique by combining the P-Median location-allocation model, for optimizing health services. First, a geodatabase, containing public hospitals, road networks and population districts, was prepared. Next, we investigated the location and services of five public hospitals in Jeddah city of KSA, by using a MCDM model that included a fuzzy overlay technique with a location-allocation model. The results showed that the allocated five hospitals served 94 out of 110 districts in the study area. Our results suggested additional hospitals must be added to ensure that the entire city is covered with timely hospital services. To improve the existing situation, we prioritized demand locations using the maximize coverage (MC) location problem model. We then used the P-Median function to find the optimal locations of hospitals, and then combined these two methods to create the MC-P-Median optimizer. This optimizer eliminated any unallocated or redundant information. Health planners can use this model to determine the best locations for public hospitals in Jeddah city and similar settings.

Mathematical problems for designing a short-distance relief goods distribution network in urban areas

In the humanitarian relief efforts after an earthquake strikes, local capabilities are critical in the first hours to assess the situation and deliver relief supplies to the victims. Usually, the relief goods required by victims are unlikely to be found in the impacted areas, and resources from other places need time to be mobilized to the affected areas. Often in earthquakes in cities in developing countries, relief goods do not arrive on time causing victims to suffer even more. In this paper, we present the mathematical formulations and solutions techniques for designing a short-distance distribution network of relief goods, which can be managed by local governments. In the preparation phase, as a part of the immediate response plan based on scenarios, the facilities location and the preliminary distribution paths must be determined. Hence, a p-median problem and a multi-depot heterogeneous fleet vehicle routing problem (MDHFVRP) must be solved. These problems are usually small and there is time to find a solution. However, in the response phase, a dynamical VRP must be solved.