Stochastic Facility Location Problem Research Articles

Fairness in accessibility of public service facilities

Our paper studies a fair stochastic facility location problem with congestion under the max-min principle. We analyze the price of fairness in a two-location problem and develop a tractable optimization framework for the general multi-location setting. Evaluating the fair solution against the utilitarian solution on Buffalo's demographic data reveals that implementing a fair solution can substantially improve fairness measures (up to 98%) with relatively limited impact on the overall service quality.

Comparative study of robust optimization model of multimaterial emergency logistics in an uncertain environment

AbstractGiven the significance of objective functions in alleviating human suffering within the humanitarian relief supply chain, it is imperative to consider both logistics costs and the cost of deprivation as ideal targets. This study investigates a multimaterial single‐cycle stochastic mixed integer model, integrating uncertainties about post‐disaster demand, supply survival, and road networks. An innovative heuristic approach is used to solve the stochastic facility location problem, which minimizes unfulfilled requests and response times in lexicographic order. This research also compares the cost‐minimization model of the pre‐positioning problem to an alternative model that maximizes demand satisfaction under resource constraints. Empirical validation through numerous case studies and solutions confirms the effectiveness and feasibility of the alternative model. The results of our experiments consistently demonstrate that the best solution provided by the alternative model not only satisfies the minimum cost model but also surpasses the conventional problem, without compromising performance. Consequently, it is crucial to avoid the cost‐minimization approach at the expense of human suffering.

Ensemble Variance Reduction Methods for Stochastic Mixed-Integer Programming and their Application to the Stochastic Facility Location Problem

Sample average approximation (SAA), the standard approach to stochastic mixed-integer programming, does not provide guidance for cases with limited computational budgets. In such settings, variance reduction is critical in identifying good decisions. This paper explores two closely related ensemble methods to determine effective decisions with a probabilistic guarantee. (a) The first approach recommends a decision by coordinating aggregation in the space of decisions as well as aggregation of objective values. This combination of aggregation methods generalizes the bagging method and the “compromise decision” of stochastic linear programming. Combining these concepts, we propose a stopping rule that provides an upper bound on the probability of early termination. (b) The second approach applies efficient computational budget allocation for objective function evaluation and contributes to identifying the best solution with a predicted lower bound on the probability of correct selection. It also reduces the variance of the upper bound estimate at optimality. Furthermore, it adaptively selects the evaluation sample size. Both approaches provide approximately optimal solutions even in cases with a huge number of scenarios, especially when scenarios are generated by using oracles/simulators. Finally, we demonstrate the effectiveness of these methods via extensive computational results for “megascale” (extremely large scale) stochastic facility location problems. History: Accepted by Pascal Van Hentenryck, Area Editor for Computational Modeling: Methods & Analysis. Funding: This work was supported by The Office of Naval Research [Grant N00014-20-1-2077] and the Air Force Office of Scientific Research [Grant FA9550-20-1-0006]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2021.0324 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2021.0324 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Code and Data Repository for Ensemble Variance Reduction Methods for Stochastic Mixed-Integer Programming and their Application to the Stochastic Facility Location Problem

Code and Data Repository for Ensemble Variance Reduction Methods for Stochastic Mixed-Integer Programming and their Application to the Stochastic Facility Location Problem

Two-stage stochastic facility location problem with disruptions and restricted shortages

In this study, we consider a reliable facility location problem where an uncertain disruptive event might affect the demand of the nodes, the allocation costs, and the available capacities at the facilities. We assume that the decision maker decides the locations and the types (size and reliability levels) of the facilities under uncertainty, while the allocation and shortage decisions are made in response to the realizations of the uncertain parameters. Motivated from a humanitarian facility location problem, we enforce an upper bound on the risk of shortages under any scenario using the popular risk measure conditional value-at-risk (CVaR). We formulate the problem as a two-stage stochastic programming model, and solve it using the L-Shaped method accelerated with additional enhancements. Our computational experiments show the value of our modeling scheme, and the effectiveness of our solution method.

Reducing disparities in transportation distance in a stochastic facility location problem

This paper focuses on improving equity in a well-known stochastic facility location problem (FLP). Specifically, given sets of potential facility locations and customers with random demand, traditional stochastic optimization models for the FLP seek to determine the number and location of facilities to open that minimize the total cost, including the fixed cost of opening facilities, expected transportation cost from customers to facilities, and expected unmet demand cost. We show that such models lead to inequitable distribution of facilities across different regions and disparities in transportation distance needed to reach those facilities. To mitigate such disparities, we first analyze a set of measures to gauge the level of equity of location decisions. Then, to investigate the impact of the method of modeling uncertainty on equity, we formulate and analyze stochastic programming (SP) and distributionally robust optimization (DRO) models that employ these measures to minimize disparities in transportation distance needed to access open facilities, assuming known and unknown distributions of the demand. Finally, we use a case study based on the Lehigh region in Pennsylvania to compare the impact of location decisions obtained using the traditional and the proposed models on equity and efficiency. Our results demonstrate how different methods of modeling equity under uncertainty can result in different location decisions with varying impacts on equity and efficiency. Notably, while prior literature shows that equity-neutral DRO models for the FLP yield robust location decisions, our results additionally demonstrate that the proposed DRO models produce equitable distributions of facilities and more effectively mitigate disparities in transportation distance than their SP and equity-neutral counterparts.

Robust Stochastic Facility Location: Sensitivity Analysis and Exact Solution

This work focuses on a broad class of facility location problems in the context of adaptive robust stochastic optimization under the state-dependent demand uncertainty. The demand is assumed to be significantly affected by related state information, such as the seasonal or socio-economic information. In particular, a state-wise ambiguity set is adopted for modeling the distributional uncertainty associated with the demand in different states. The conditional distributional characteristics in each state are described by a support, as well as by mean and dispersion measures, which are assumed to be conic representable. A robust sensitivity analysis is performed, in which, on the one hand, we analyze the impact of the change in ambiguity-set parameters (e.g., state probabilities, mean value abounds, and dispersion bounds in different states) onto the optimal worst-case expected total cost using the ambiguity dual variables. On the other hand, we analyze the impact of the change in location design onto the worst-case expected second-stage cost and show that the sensitivity bounds are fully described as the worst-case expected shadow-capacity cost. As for the solution approach, we propose a nested Benders decomposition algorithm for solving the model exactly, which leverages the subgradients of the worst-case expected second-stage cost at the location decisions formed insightfully by the associated worst-case distributions. The nested Benders decomposition approach ensures a finite-step convergence, which can also be regarded as an extension of the classic L-shaped algorithm for two-stage stochastic programming to our state-wise, robust stochastic facility location problem with conic representable ambiguity. Finally, the results of a series of numerical experiments are presented that justify the value of the state-wise distributional information incorporated in our robust stochastic facility location model, the robustness of the model, and the performance of the exact solution approach.

A branch-and-Benders-cut algorithm for a bi-objective stochastic facility location problem

In many real-world optimization problems, more than one objective plays a role and input parameters are subject to uncertainty. In this paper, motivated by applications in disaster relief and public facility location, we model and solve a bi-objective stochastic facility location problem. The considered objectives are cost and covered demand, where the demand at the different population centers is uncertain but its probability distribution is known. The latter information is used to produce a set of scenarios. In order to solve the underlying optimization problem, we apply a Benders’ type decomposition approach which is known as the L-shaped method for stochastic programming and we embed it into a recently developed branch-and-bound framework for bi-objective integer optimization. We analyze and compare different cut generation schemes and we show how they affect lower bound set computations, so as to identify the best performing approach. Finally, we compare the branch-and-Benders-cut approach to a straight-forward branch-and-bound implementation based on the deterministic equivalent formulation.

A matheuristic for the stochastic facility location problem

In this paper, we describe a matheuristic to solve the stochastic facility location problem which determines the location and size of storage facilities, the quantities of various types of supplies stored in each facility, and the assignment of demand locations to the open facilities, which minimize unmet demand and response time in lexicographic order. We assume uncertainties about demands, inventory spoilage, and transportation network availability. A good example where such a formulation makes sense is the the problem of pre-positioning emergency supplies, which aims to increase disaster preparedness by making the relief items readily available to people in need. The matheuristic employs iterated local search techniques to look for good location and inventory configurations, and uses CPLEX to optimize the assignments. Numerical experiments on a number of case studies and random instances for the pre-positioning problem demonstrate the effectiveness and efficiency of the matheuristic, which is shown to be particularly useful for tackling larger instances that are intractable for exact solvers. The matheuristic is therefore a contribution to the literature on heuristic approaches to solving facility location under uncertainties, can be used to further study the particular variant of the facility location problem, and can also support humanitarian logisticians in their planning of pre-positioning strategies.

Stochastic facility location problem in a competitive situation: A game theory model for emergency financial services

Facility location problems under conditions of uncertainty involve various situations, such as industrial plants, warehouses, distribution centers, and emergency services (ambulances, police cars)....

Approximation algorithm for squared metric two-stage stochastic facility location problem

In this paper, we consider a variant of the classical uncapacitated facility location problem, so-called squared metric two-stage stochastic facility location problem (SM-2-SFLP) which can treat the uncertainty of the set of clients and facility costs. We assume that the connection cost is squared metric, a variant of the metric case which is widely researched. We give a new 0–1 integer linear programming for SM-2-SFLP. Based on the new formulation, we apply two known algorithms to SM-2-SFLP, and analyze the approximation ratio and per-scenario bound respectively.

Effects of feasibility cuts in Lagrangian relaxation for a two-stage stochastic facility location and network flow problem

This paper studies a two-stage stochastic facility location and network flow problem with uncertainty in demand, supply, network availability, transportation costs, and cost of facility activation. The goal is to design a minimum cost long-term network of facilities, in anticipation of uncertain supply, demand, and network structure. Facility location decisions are made in the first stage and facility activation and network flow decisions are made in the second stage. We develop a branch and price algorithm built on a Lagrangian relaxation with two subproblems, one per stage of decision making. To improve convergence, we show that the Lagrangian subproblems could be strengthened by using Benders decomposition. Namely, we add to the first stage Lagrangian subproblem, Benders feasibility cuts generated from the second stage Lagrangian subproblem, and guarantee that the former only generates optimality cuts. Numerical results show that feasibility cuts tighten the Lagrangian duality gap in the root node of the branch and price tree, and solutions generated by our algorithm improve over the literature in terms of computational time and number of solved instances.

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

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

An edge-based stochastic facility location problem in UAV-supported humanitarian relief logistics: a case study of Tehran earthquake

The application of facility location problems in choosing the best location of relief distribution centers plays a salient role in emergency operations of large-scale disasters. On the premise that the service recipients are uniformly distributed along the network edges, this study investigates a combined mobile and immobile pre-earthquake facility location problem. A predefined number of locations are to be selected among a set of potential locations. Each facility is used in the relief distribution operation. It is incontrovertible that due to earthquakes, some network edges collapse and corresponding areas may lose their accessibility. Thus in this study, it is assumed that people on intact and accessible edges travel to the location of the distribution centers to receive the relief. For those who are located on collapsed or inaccessible network edges, the medium-scale unmanned aerial vehicle (UAV) helicopters are utilized in the relief distribution operation. This study aims to develop a mathematical model which minimizes the aggregate traveling time for both people and UAVs over a set of feasible scenarios. Since the network problems are NP-hard, some metaheuristic algorithms are developed to solve the proposed model. In order to demonstrate the applicability of developed model, a case study based on feasible earthquake scenarios in Tehran is presented.

Approximation algorithms for k-level stochastic facility location problems

In the k-level facility location problem (FLP), we are given a set of facilities, each associated with one of k levels, and a set of clients. We have to connect each client to a chain of opened facilities spanning all levels, minimizing the sum of opening and connection costs. This paper considers the k-level stochastic FLP, with two stages, when the set of clients is only known in the second stage. There is a set of scenarios, each occurring with a given probability. A facility may be opened in any stage, however, the cost of opening a facility in the second stage depends on the realized scenario. The objective is to minimize the expected total cost. For the stage-constrained variant, when clients must be served by facilities opened in the same stage, we present a $$(4-o(1))$$(4-o(1))-approximation, improving on the 4-approximation by Wang et al. (Oper Res Lett 39(2):160---161, 2011) for each k. In the case with $$k=2,\,3$$k=2,3, the algorithm achieves factors 2.56 and 2.78, resp., which improves the $$(3+\epsilon )$$(3+∈)-approximation for $$k=2$$k=2 by Wu et al. (Theor Comput Sci 562:213---226, 2015). For the non-stage-constrained version, we give the first approximation for the problem, achieving a factor of 3.495 for the case with $$k = 2$$k=2, and $$2k-1+o(1)$$2k-1+o(1) in general.

Evasive flow capture: A multi-period stochastic facility location problem with independent demand

We introduce the problem of locating facilities over a finite time horizon with the goal of intercepting stochastic traffic flows that exhibit evasive behavior, which arises when locating weigh-in-motion systems, tollbooths, vehicle inspection stations, or other fixed flow-capturing facilities used for law enforcement. The problem can be formulated as a multi-stage, mixed-integer stochastic program; however, under certain independence assumptions, this can be reformulated as a large two-stage stochastic program, enabling us to solve much larger instances. We additionally propose an algorithm based on Lagrangian relaxation that separates the reformulated stochastic program into a variant of a deterministic knapsack problem and a sum of time-decoupled single-period stochastic programs that can be solved independently. The model and algorithm are tested on instances involving road networks of Nevada and Vermont. A comparison with the previously studied single-period stochastic programming approach shows that the newly proposed multi-period model substantially reduces the expected cost.

Modeling and Optimization of a Spatial Detection System

Oil and gas companies are drilling and developing fields in the Arctic Ocean, which is an environment with ice floes. These companies must protect their platforms from ice floe collisions. One proposal is to use a system that consists of autonomous underwater vehicles (AUVs) and docking stations. The AUVs measure the under-water topography of the ice floes, while the docking stations launch the AUVs and recharge their batteries. Given resource constraints, we optimize locations and quantities for the docking stations and the AUVs, as well as the AUV scheduling policies, to maximize security of the platform. We model the system using a multistage stochastic facility location problem to optimize the docking station locations, the AUV allocations, and the scheduling policies of the AUVs. A two-stage stochastic facility location problem and two efficient online scheduling heuristics provide lower bounds and upper bounds for the multistage model. Even though the model is motivated by an oil industry project, most of the modeling and optimization methods apply more broadly to two-dimensional radial detection.

A hybrid algorithm for stochastic single-source capacitated facility location problem with service level requirements

A hybrid algorithm for stochastic single-source capacitated facility location problem with service level requirements

Locating and Designing a Biorefinery Supply Chain under Uncertainty in Navarre: A Stochastic Facility Location Problem Case

The need for renewable energy sources is quickly growing in order to reduce the greenhouse gas emissions. Moreover, Navarre, a European region located in Northern Spain is, currently, a global leader in the production and use of renewable energy. Actually, more than 80% of its electricity production comes from renewable sources (mainly wind and water). Then, having the purpose of increasing the renewable energy sources diversification, the region aims to locate a biorefinery plant which mainly serves Northern Spain. Locating decisions are considered strategic, immobilizing a large amount of resources and involving an important group of industrial actors. Therefore, they initially show a significant impact on investment costs, and later, on the operating costs when the facility is already running. This location activity has also important environmental influence due to the usual performance of the biorefinery, involving also the transportation and logistic activities because of the supply chain procurement. Once the biorefinery has been located, another problem arises: the design of the supply chain with its classical operational decisions: which crops are going to be harvested, when they are going to be collected and how we should transport the feedstock to the biorefinery. Apart from this, dealing with farms production is always dealing with uncertainty. Thus, climate and weather, competitors and alternate uses, are key factors which influence the availability of biomass. For that reason, uncertainty must be taken into account in order to avoid stockouts that allow us the optimization of the total expected cost. Moreover, estimated feedstock availability is crucial to determine the optimal plant size. Therefore, the results provide us not only the best location of the biorefinery from the economic point of view, but also the variation on feedstock disposal that eventually the biorefinery could intake along with its final size.

An improved per-scenario bound for the two-stage stochastic facility location problem

We study the two-stage stochastic facility location problem (2-SFLP) by proposing an LP (location problem)-rounding approximation algorithm with 2.3613 per-scenario bound for this problem, improving the previously best per-scenario bound of 2.4957.