Sherali-Adams gaps, flow-cover inequalities and generalized configurations for capacity-constrained Facility Location

Metric facility location is a well-studied problem for which linear programming methods have been used with great success in deriving approximation algorithms. The capacity-constrained generalizations, such as capacitated facility location (CFL) and lower-bounded facility location (LBFL), have proved notorious as far as LP-based approximation is concerned: while there are local-search-based constant-factor approximations, there is no known linear relaxation with constant integrality gap. According to Williamson and Shmoys devising a relaxation-based approximation for \cfl\ is among the top 10 open problems in approximation algorithms. This paper advances significantly the state-of-the-art on the effectiveness of linear programming for capacity-constrained facility location through a host of impossibility results for both CFL and LBFL. We show that the relaxations obtained from the natural LP at $\Omega(n)$ levels of the Sherali-Adams hierarchy have an unbounded gap, partially answering an open question of \cite{LiS13, AnBS13}. Here, $n$ denotes the number of facilities in the instance. Building on the ideas for this result, we prove that the standard CFL relaxation enriched with the generalized flow-cover valid inequalities \cite{AardalPW95} has also an unbounded gap. This disproves a long-standing conjecture of \cite{LeviSS12}. We finally introduce the family of proper relaxations which generalizes to its logical extreme the classic star relaxation and captures general configuration-style LPs. We characterize the behavior of proper relaxations for CFL and LBFL through a sharp threshold phenomenon.Comment: arXiv admin note: substantial text overlap with arXiv:1305.599

Healthcare Facility Location and Capacity Configuration under Stochastic Demand

This dissertation addresses two topics. The first topic is strategic dynamic supply chain reconfiguration (DSCR) problem, in which the proposed capacity configuration network is employed in the second topic: healthcare facility location and capacity configuration under stochastic demand. The second topic investigates two problems: the stochastic, single healthcare facility location and capacity configuration problem (SSHFCP) in a competitive environment and the stochastic, multiple healthcare facility location and capacity configuration problem (SMHFCP) based on a location-allocation model. The DSCR problem is to prescribe the location and capacity of each facility, select links used for transportation, and plan material flows through the supply chain, including production, inventory, backorder, and outsourcing levels. The objective is to minimize total cost. The network must be dynamically reconfigured (i.e., by opening facilities, expanding and/or contracting their capacities, and closing facilities) over time to accommodate changing trends in demand and/or costs. This research proposes a network-based model of DSCR and compares it with a traditional mixed integer programming (MIP) formulation via extensive, large-scale computational tests and sensitivity analyses, showing that the network-based model offers superior solvability. The SSHFCP is to prescribe the location and multi-service, multi-period capacity configuration of facility facing competition from existing facilities under uncertain patient demand, so that the expected excess revenue (i.e., the amount by which revenue exceeds cost) of the new facility is maximized. This dissertation describes a solution methodology that relates practical features relative to healthcare, including a multiplicative competitive interaction (MCI) model to reflect competition among providers and a method to model the stochastic problem as a deterministic resource constrained shortest path problem (RCSPP) on a specially constructed network, which can be solved in pseudo-polynomial time. This dissertation proposes two solution methods to SMHFCP. The dissertation shows that first method, a column-generation heuristic, solves test instances to near optimality; and the second one, an approximation method, provides a fast runtime with a bounding procedure to assess the quality of a solution. The application of SSHFCP and SMHFCP in locating and configuring new primary care centers in mid-Texas rural area validates the real business decision of industrial collaborators

A regret model applied to the facility location problem with limited capacity facilities

This article addresses issues related to location and allocation problems. Herein, we intend to demonstrate the influence of congestion, through the random number generation, of such systems in final solutions. An algorithm is presented which, in addition to the GRASP, incorporates the Regret with the pminmax method to evaluate the heuristic solution obtained with regard to its robustness for different scenarios. Taking as our point of departure the Facility Location Problem proposed by Balinski [27], an alternative perspective is added associating regret values to particular solutions.N/

Assessing performance in health care: A mathematical programming approach for the re-design of primary health care networks

Mathematical models allow studying complex systems. In particular, optimal facility location models provide a sound framework to assess the performance of first-level of health care networks. In this work, a methodology founded on need/offer/demand quantification through a facility location-based mathematical model is proposed to assess the performance of existing networks of Primary Health Care Centers (PHCC) and assist in its re-design. The proposed re-design problem investigates the re-allocation of existing resources within the given infrastructure (existing PHCCs) to better satisfy the estimated health needs of the target population. This problem has not been widely addressed in the open literature despite its paramount importance in modern societies with fast demographic dynamics and constrained investment capacities. The model seeks to optimally assign the required type of service and the corresponding capacity to each PHCC (offer). The objective function to be maximized is the number of (needed) patients’ visits effectively covered by the network (demand). The following constraints are explicitly considered: i) geographic accessibility from need centers to PHCCs, ii) maximum delivery capacity of each service in each PHCC, and iii) total budget regarding fixed, variable, and relocation costs. The proposed methodology was applied to a medium-size city. Results show that the non-attended necessity can be reduced by introducing capacity modifications in the existing network. Moreover, different solutions are obtained if budgetary restrictions or minimum attention volume constraints are included. This reveals how model-based decision support tools can help health decision-makers assessing primary health care network performance.Fil: Elorza, Maria Eugenia. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Investigaciones Económicas y Sociales del Sur. Universidad Nacional del Sur. Departamento de Economía. Instituto de Investigaciones Económicas y Sociales del Sur; ArgentinaFil: Moscoso, Nebel Silvana. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Investigaciones Económicas y Sociales del Sur. Universidad Nacional del Sur. Departamento de Economía. Instituto de Investigaciones Económicas y Sociales del Sur; ArgentinaFil: Blanco, Anibal Manuel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Planta Piloto de Ingeniería Química. Universidad Nacional del Sur. Planta Piloto de Ingeniería Química; Argentin