Sample Average Approximation Algorithm Research Articles

Multi-period reverse logistics network design for water resource management in hydraulic fracturing

This paper addresses the effective management of water resources during the hydraulic fracturing process used for natural gas extraction. With increasing demand and scrutiny on shale gas production due to economic, environmental, and social factors, effective water management strategies are needed to address the challenges of flow-back water usage during hydraulic fracturing. This research contributes to the existing literature from a multi-period reverse logistics network design perspective with realistic modeling approaches. A mathematical formulation is developed for optimizing the design of a multi-period reverse logistics network for effective water resource management that incorporates production scheduling of a hydraulic fracturing process. This mixed integer programming model determines the optimal locations and capacities of impoundments, treatment facilities, and disposal sites while minimizing fixed and operational costs under constraints such as freshwater withdrawal limits, production schedules, inventory balance, and treatment and disposal options. A two-stage stochastic programming formulation is further developed as a means to address the uncertainty associated with water flows during the fracturing operation. The strategic location and capacity decisions are given in the first stage of this formulation, whereas operational decisions such as water flows are handled in the second stage under independently sampled scenarios. A sample average approximation algorithm is proposed to solve the stochastic formulation. The model is tested on a case study from the United States that explores potential impacts on water resource management due to cost variations, changes in operational parameters, and uncertainty in demands and supplies. The base case setup delivers a 12.8% reduction in freshwater usage. The benefit of multi-period reverse logistics network design is further demonstrated through variations in water demand and equipment degradation. The managerial insights derived through sensitivity analysis highlight the values of the proposed multi-period formulation, capacity expansions, and parameter estimations.

Assignment of Freight Traffic in a Large-scale Intermodal Network under Uncertainty

This paper presents a methodology for freight traffic assignment in a large-scale road-rail intermodal network under uncertainty. Network uncertainties caused by natural disasters have dramatically increased in recent years. Several of these disasters (e.g., Hurricane Sandy, Mississippi River Flooding, and Hurricane Harvey) severely disrupted the U.S. freight transportation network, and consequently, the supply chain. To account for these network uncertainties, a stochastic freight traffic assignment model is formulated. An algorithmic framework, involving the sample average approximation and gradient projection algorithm, is proposed to solve this challenging problem. The developed methodology is tested on the U.S. intermodal network with freight flow data from the Freight Analysis Framework. The experiments consider three types of natural disasters that have different risks and impacts on transportation networks: earthquakes, hurricanes, and floods. It is found that for all disaster scenarios, freight ton-miles are higher compared to the base case without uncertainty. The increase in freight ton-miles is the highest under the flooding scenario; this is because there are more states in the flood-risk areas, and they are scattered throughout the U.S.

Transmission Defense Hardening Against Typhoon Disasters Under Decision-Dependent Uncertainty

Transmission defense hardening (TDH) is an effective practice to ensure the safe operation of the power system during and after natural disasters, such as typhoons. However, the coupling among the typhoon disaster, hardening decision, and transmission line status is not addressed in the previous research works. Besides, the enhancement measures for different types of transmission line failures have not yet been assessed. This paper proposes a comprehensive framework to improve power transmission system resilience against typhoon disasters. Firstly, the motion path and the wind field of typhoons are simulated using Monte Carlo sampling to quantify the spatiotemporal impacts of wind speed on the transmission line status. The decision-dependent uncertainty (DDU), which reflects the effects of hardening decisions on the transmission lines, is addressed by sampling according to the corresponding failure probabilities. Then, the scenario simulation is integrated into the proposed two-stage stochastic mixed-integer programming (SMIP) model considering various enhancement measures for different failures. Since the way to handle the DDU needs a large number of scenarios, it leads to high computation complexity. Hence, the sample average approximation (SAA) algorithm is introduced to cope with it. Finally, the proposed framework and its solution algorithm are carried out on the modified IEEE RTS-79 system and the modified IEEE 118-Bus system. The significant cost saving and resilience improvement demonstrate the effectiveness of the proposed framework.

Solving the stochastic team orienteering problem: comparing simheuristics with the sample average approximation method

AbstractThe team orienteering problem (TOP) is an NP‐hard optimization problem with an increasing number of potential applications in smart cities, humanitarian logistics, wildfire surveillance, etc. In the TOP, a fixed fleet of vehicles is employed to obtain rewards by visiting nodes in a network. All vehicles share common origin and destination locations. Since each vehicle has a limitation in time or traveling distance, not all nodes in the network can be visited. Hence, the goal is focused on the maximization of the collected reward, taking into account the aforementioned constraints. Most of the existing literature on the TOP focuses on its deterministic version, where rewards and travel times are assumed to be predefined values. This paper focuses on a more realistic TOP version, where travel times are modeled as random variables, which introduces reliability issues in the solutions due to the route‐length constraint. In order to deal with these complexities, we propose a simheuristic algorithm that hybridizes biased‐randomized heuristics with a variable neighborhood search and MCS. To test the quality of the solutions generated by the proposed simheuristic approach, we employ the well‐known sample average approximation (SAA) method, as well as a combination model that hybridizes the metaheuristic used in the simheuristic approach with the SAA algorithm. The results show that our proposed simheuristic outperforms the SAA and the hybrid model both on the objective function values and computational time.

How Big Should Your Data Really Be? Data-Driven Newsvendor: Learning One Sample at a Time

We study the classical newsvendor problem in which the decision maker must trade off underage and overage costs. In contrast to the typical setting, we assume that the decision maker does not know the underlying distribution driving uncertainty but has only access to historical data. In turn, the key questions are how to map existing data to a decision and what type of performance to expect as a function of the data size. We analyze the classical setting with access to past samples drawn from the distribution (e.g., past demand), focusing not only on asymptotic performance but also on what we call the transient regime of learning, that is, performance for arbitrary data sizes. We evaluate the performance of any algorithm through its worst-case relative expected regret, compared with an oracle with knowledge of the distribution. We provide the first finite sample exact analysis of the classical sample average approximation (SAA) algorithm for this class of problems across all data sizes. This allows one to uncover novel fundamental insights on the value of data: It reveals that tens of samples are sufficient to perform very efficiently but also that more data can lead to worse out-of-sample performance for SAA. We then focus on the general class of mappings from data to decisions without any restriction on the set of policies and derive an optimal algorithm (in the minimax sense) and characterize its associated performance. This leads to significant improvements for limited data sizes and allows to exactly quantify the value of historical information. This paper was accepted by David Simchi-Levi, data science. Supplemental Material: The data files and online appendix are available at https://doi.org/10.1287/mnsc.2023.4725 .

Disassembly leveling and lot-sizing for multiple product types with uncertain component demands

Disassembly leveling and lot-sizing is an integrated problem of determining the depth of disassembling a product and the resulting amounts of disassembling the product and its subassemblies to satisfy component demands. This study addresses two stochastic versions of the problem for multiple product types with uncertain component demands: a basic problem and its extension with parts commonality. The objective is to minimize the sum of expected setup and operation costs. For the basic problem without parts commonality, an optimal sample average approximation algorithm is proposed and then illustrated with a numerical example after formulating it as a stochastic integer programing model. Also, due to the complexity of the extended problem, sample average approximation based heuristics are proposed and computational results are reported. In particular, the results show that the best heuristic gives near optimal solutions for small sized test instances.

Integrated consultation and chemotherapy scheduling with stochastic treatment times

This paper studies the integrated scheduling of consultation and treatment appointments for chemotherapy patients, while taking into account the stochastic duration of injection. Patients may require one or both types of consultation and treatment appointments. The objective is to minimize the clinic’s overtime and the waiting time of patients in the clinic. To formulate the problem, we develop two two-stage stochastic programming models. We also propose a sample average approximation algorithm as the solution method. To improve the efficiency of our solution approach, we devise a specialized algorithm that quickly evaluates a given first-stage solution for a large set of scenarios, without solving the second-stage models. Several computational experiments are carried out to evaluate the performance of proposed models and algorithms.

AWS IoT analytics platform for microgrid operation management

Microgrid (MG) represents a promising opportunity for integrating renewable energy systems with the electric power grid. However, numerous complexities need to be addressed in the process. The electrical grid is complex, vulnerable, and centralized. Thus, the integration is challenging owing to the stochastic nature of renewable energy generation, which affects the possibility of reliable forecasting. The wastage due to poor estimation of clean energy generation discourages new investments in this area. However, recent advancements in big data technologies enable processing a large amount of data captured from multiple sources in real-time. It opens the possibility of improving the operational optimization of MGs and the performance of forecasting models. The overall MG problem is formulated using a two-stage stochastic mixed-integer linear programming problem with recourse. Amazon Web Services (AWS) IoT analytics platform inputs data in real-time and runs a sophisticated wind generation forecast analysis. The stochastic model is solved using the Sample Average Approximation (SAA) algorithm. The innovative methodology leads to significant improvements in the total average operating cost by integrating AWS IoT Analytics compared to traditional methods that use historical data. Computations are performed for different power-grid settings, including a power-outage, and different power generators units capacities with total operation average cost savings of 7.6% and 5.9%, respectively. Sensitivity analysis showed that the SSA algorithm could solve all the instances by providing high-quality solutions. The AWS IoT strategy outperformed at 7.7 % and 3.6% for optimality gap and CPU time, respectively. We examined an actual case in Peru for an agricultural application to assess the performance of a stochastic optimization model with a real-time IoT wind generation forecast strategy. The results revealed the following capabilities of our novel framework: (1) it can realize higher cost savings from the MG operating systems; (2) improve real-time renewable energy forecasting; (3) facilitate robust decision-making under conditions of uncertainty.

Staff Planning for Hospitals with Implicit Cost Estimation and Stochastic Optimization

We consider the anesthesiologist staff planning problem for operating services departments in large multi‐specialty hospitals without limit on anesthesiologist supply, where the planner makes monthly and daily decisions to minimize total costs. Each month the staff planner decides the number of anesthesiologists on regular duty and an on‐call consideration list for each day of the following month. In addition, each day, the staff planner decides how many on‐call anesthesiologists to call for the following day. Total costs consist of explicit and implicit costs. Explicit costs include the costs of calling an anesthesiologist and overtime costs. These costs are specified by the organization. Implicit costs encompass costs of not calling an on‐call anesthesiologist and under‐utilizing an anesthesiologist, and these have to be deduced from past decisions. We model the staff planning problem as a two‐stage integer stochastic dynamic program. We develop structural properties of this model and use them in a sample average approximation algorithm constructed to solve this problem. We also develop a procedure to estimate the implicit costs, which are included in this model. Using data from the operating services department at the UCLA Ronald Reagan Medical Center, our model shows the potential to reduce overall costs by 16%. We provide managerial insights related to the relative scale of these costs, hiring decisions by service, sensitivity to cost parameters, and improvements in the prediction of the booked time durations.

Cash versus in-kind transfer programs in humanitarian operations: An optimization program and a case study

Humanitarian organizations, mandated with responding to emergencies, generally provide food assistance via in-kind and/or cash transfers. Although cash and in-kind transfers have had varying effects across different regions, the superiority of one over the other has been debated. This study considers a mixed strategy that includes both cash and in-kind transfer in the presence of finite budget, given a slow-onset disaster such as famine or drought. Importantly, it proposes an evidence-based framework based on the given data. Specifically, a two-stage stochastic program with recourse is proposed, where uncertainty stems from the slow-onset disaster that has non-uniform impact across a given geographical region. The proposed program is first used to study the slow-onset disaster situation in Kenya, and then to evaluate the performance of the cash versus in-kind transfer programs. We also solved larger size problem instances using the sample average approximation (SAA) algorithm, and the resulting analyses underscore the deductions of the case study that although cash transfer is more efficient than in-kind transfer, however, the latter is inevitable due to local unavailability of certain commodities.

A Stochastic Distribution System Market Clearing and Settlement Model With Distributed Renewable Energy Utilization Constraints

The increasing integration of distributed generation (DG) has changed the way modern power systems are planned and operated. However, new market mechanisms at the retail level are still needed to effectively integrate energy resources and emerging market participants in the distribution grid. In this article, a stochastic market clearing and settlement model with distributed renewable energy (DRE) utilization constraints is presented. The proposed model considers the uncertainties associated with the DRE productions in the distribution grid and ensures that a certain portion of the total available DRE at a specified time will be utilized with a high probability. The problem is formulated as a chance-constrained two-stage stochastic programming model that minimizes the expected distribution system operation cost and is solved through a sample average approximation algorithm. Case studies are conducted to verify the effectiveness of the proposed model for different DRE utilization requirements.

Integrated stochastic disassembly line balancing and planning problem with machine specificity

The disassembly is a fundamental basis in converting End-of-Life (EOL) products into useful components. Related research becomes popular recently due to the increasing awareness of environmental protection and energy conservation. Yet, there are many opening questions needed to be investigated, especially the efficient coordination of different-level decisions under uncertainty is a big challenge. In this paper, a novel integrated stochastic disassembly line balancing and planning problem is studied to minimise the system cost, where component yield ratios and demands are assumed to be uncertain. In this work, machine specificities are considered for task processing, such as price, ability, and capacity. For the problem, a two-stage non-linear stochastic programming model is first constructed. Then, it is further transformed into a linear formulation. Based on problem property analysis, a valid inequality is proposed to reduce the search space of optimal solutions. Finally, a sample average approximation (SAA) and an L-shaped algorithm are adopted to solve the problem. Numerical experiments on randomly generated instances demonstrate that the valid inequality can save around 11% of average computation time, and the L-shaped algorithm can save around 64% of average computation time compared with the SAA algorithm without a big sacrifice of the solution quality.

How Big Should Your Data Really Be? Data-Driven Newsvendor and the Transient of Learning

We study the classical newsvendor problem in which the decision-maker must trade-off underage and overage costs. In contrast to the typical setting, we assume that the decision- maker does not know the underlying distribution driving uncertainty but has only access to historical data. In turn, the key questions are how to map existing data to a decision and what type of performance to expect as a function of the data size. We analyze the classical setting with access to past samples drawn from the distribution (e.g., past demand), focusing not only on asymptotic performance but also on what we call the transient of learning, i.e., performance for arbitrary data sizes. We evaluate the performance of any algorithm through its worst-case relative expected regret, compared to an oracle with knowledge of the distribution. We provide the first finite sample exact analysis of the classical Sample Average Approximation (SAA) algorithm for this class of problems across all data sizes. This allows to uncover novel fundamental insights on the value of data: it reveals that tens of samples are sufficient to perform very efficiently but also that more data can lead to worse out-of-sample performance for SAA. We then focus on the general class of mappings from data to decisions without any restriction on the set of policies and derive an optimal algorithm as well as characterize its associated performance. This leads to significant improvements for limited data sizes, and allows to exactly quantify the value of historical information.

Energy-Aware Application Placement in Mobile Edge Computing: A Stochastic Optimization Approach

The Quality of Service (QoS) in Mobile Edge Computing (MEC) systems is significantly dependent on the application offloading and placement decisions. Due to the movement of users in MEC networks, an optimal application placement might turn into the least efficient placement in few minutes. Thus, it is crucial to take the dynamics of the system into account when designing application placement mechanisms. On the other hand, energy consumption of servers is a significant component of the cost of services in MEC systems and must also be considered in the design of the mechanisms. In this article, we model the problem of energy-aware application placement in edge computing systems as a multi-stage stochastic program. The objective is to maximize the QoS of the system while taking into account the limited energy budget of the edge servers. To solve the problem, we design a novel parallel Sample Average Approximation (SAA) algorithm. We conduct an extensive experimental analysis to evaluate the performance of the proposed algorithm using real-world trace data.

Stochastic Runway Scheduling Problem With Partial Distribution Information of Random Parameters

Air traffic flow management is one of the most important operations in terminal airports heavily relying on advanced intelligence transportation techniques. This work considers a two-stage runway scheduling problem given a set of flights with uncertain arrival times. The first-stage problem is to identify a sequence of aircraft weight classes (e.g., Heavy, Large and Small) that minimizes runway occupying time (i.e., makespan). Then the second-stage decision is dedicated to scheduling the flights as punctually as possible after their arrival times realized, which translates into determining a sequence of flights for each aircraft category such that the total deviation time imposed on the flights is minimized. Instead of an exactly known probability distribution, information on uncertain parameters is limited (i.e., ambiguous), such as means, mean absolute deviations and support set of random parameters derived from historical data. Under this information on the random parameters, an ambiguous mixed-integer stochastic optimization model is proposed. For such a problem, we approximately construct a worst-case discrete probability distribution with three possible realizations per random parameter, and adopt a hybrid sample average approximation algorithm in which genetic algorithms are used to replace commercial solvers. To illustrate the effectiveness and efficiency of the proposed model and algorithm, extensive numerical experiments are carried out.

Determining the Optimal Location of Vehicle Inspection Facilities Under Uncertainty via New Optimization Approaches

This study addresses the problem of optimally locating vehicle inspection facilities under uncertain customer demand and varying velocity considering regional constraints. The objective is to simultaneously minimize the transportation time of all customers and their transportation cost, while ensuring consumers to reach their desired destinations within their expected time and cost. We study two variants of the problem: vehicle inspection location with complete probability distributions of customer demand and vehicle velocity, and that with partial information of customer demand and vehicle velocity, i.e., only the supports and means of the stochastic variables are known. For the former problem, an expected value model and a chance-constrained program are first formulated. Then based on explored problem properties, the expected value program is equivalently reformulated as a deterministic non-linear program. To efficiently deal with the chance-constrained program, a sample average approximation (SAA)-based approach is proposed. For the latter one, we develop a new distribution-free model. Computational results for benchmark examples demonstrate that: i) for the former, the proposed deterministic program reformulation-based approach and SAA algorithm outperform the state-of-the-art approaches; ii) for the latter, the proposed distribution-free model can effectively deal with the problem with partial demand and velocity information.

Integrating decisions with advance supply information in an assemble‐to‐order system

AbstractWe study a periodic‐review assemble‐to‐order (ATO) system with multiple components and multiple products, in which the inventory replenishment for each component follows an independent base‐stock policy and stochastic product demands are satisfied according to a First‐Come‐First‐Served rule. We assume that the replenishment for various component suffers from lead time uncertainty. However, the decision maker has the so‐called advance supply information (ASI) associated with the lead times and thus can take advantage of the information for system optimization. We propose a multistage stochastic integer program that incorporates ASI to address the joint optimization of inventory replenishment and component allocation. The optimal base‐stock policy for the inventory replenishment is determined using the sample average approximation algorithm. Also, we provide a modified order‐based component allocation (MOBCA) heuristic for the component allocation. We additionally consider a special case of the variable lead times where the resulting two‐stage stochastic programming model can be characterized as a single‐scenario case of the proposed multistage model. We carry out extensive computational studies to quantify the benefits of integrating ASI into joint optimization and to explore the possibility of employing the two‐stage model as a relatively efficient approximation scheme for the multistage model.

Development of Sustainable Recycling Investment Framework Considering Uncertain Demand and Nonlinear Recycling Cost

This paper presents a more active and efficient recycling investment strategy that considers the balances among the current production constraints, manufacturing profits, and recycling investments for a sustainable circular economy as compared to the current methods. While existing production planning has numerous uncertainties and nonlinear characteristics, the circular economy-based production planning constitutes more complex uncertainties and nonlinear characteristics that result from an uncertain return rate, demand uncertainties, and nonlinear return on investment costs. This paper suggests a stochastic nonlinear programming model-based active recycling investment framework so as to generate a more effective process plan to handle these characteristics. In the proposed framework, recycling investment strategies are quantitatively analyzed when considering uncertain demand and unclear production conditions. In addition, the effective solving techniques for the circular economy based production framework are obtained while using Monte-Carlo based sample average approximation and memetic algorithm. To prove the effectiveness of the proposed framework, it is implemented for a given system and the numerical analyses that were conducted for the various sustainable manufacturing scenarios.

A Bilateral Tradeoff Decision Model for Wind Power Utilization with Extensive Load Scheduling

In this paper, we present the extensive load scheduling problem with intermittent and uncertain wind power availability. A chance-constrained bilateral tradeoff decision model is established to solve the problem. Our model aims at maximizing the wind power utilization and minimizing the system operation cost simultaneously by means of responsive loads, which are precisely divided into shiftable loads and high-energy loads. The chance constraint is applied to restrict the system imbalance with a small probability. Then, a revised sample average approximation (SAA) algorithm is developed to transform the chance constraint into sample average reformulations. Furthermore, the multi-objective differential evolution (MODE) method combined with SAA is proposed to solve the problem. Experiments enabling an effectiveness analysis of the two kinds of responsive loads are performed on the power system in Yancheng. The research of parameters of MODE, the sensitivity of different risk levels and the influence of iteration numbers are discussed. Finally, computational results prove that the combination of shiftable loads and high-energy loads have a better effect than adopting shiftable loads and high-energy loads separately, and the proposed method is convergent and valid in solving the problem.

Pre-positioning of relief items under road/facility vulnerability with concurrent restoration and relief transportation

Planning for response to sudden-onset disasters such as earthquakes, hurricanes, or floods needs to take into account the inherent uncertainties regarding the disaster and its impacts on the affected people as well as the logistics network. This article focuses on the design of a multi-echelon humanitarian response network, where the pre-disaster decisions of warehouse location and item pre-positioning are subject to uncertainties in relief item demand and vulnerability of roads and facilities following the disaster. Once the disaster strikes, relief transportation is accompanied by simultaneous repair of blocked roads, which delays the transportation process, but gradually increases the connectivity of the network at the same time. A two-stage stochastic program is formulated to model this system and a Sample Average Approximation (SAA) scheme is proposed for its heuristic solution. To enhance the efficiency of the SAA algorithm, we introduce a number of valid inequalities and bounds on the objective value. Computational experiments on a potential earthquake scenario in Istanbul, Turkey show that the SAA scheme is able to provide an accurate approximation of the objective function in reasonable time, and can help drive policy-based implications that may be applicable in preparation for similar potential disasters.