Two-stage Stochastic Programming Research Articles

A two-stage stochastic programming approach for generation and transmission maintenance scheduling with risk management

A two-stage stochastic programming approach for generation and transmission maintenance scheduling with risk management

A risk-averse latency location-routing problem with stochastic travel times

In this paper, a latency location-routing problem with stochastic travel times is investigated. The problem is cast as a two-stage stochastic program. The ex-ante decision comprises the location of the depots. The ex-post decision regards the routing, which adapts to the observed travel times. A risk-averse decision-maker is assumed, which is conveyed by adopting the latency CVaRα as the objective function. The problem is formulated mathematically. An efficient multi-start variable neighborhood search algorithm is proposed for tackling the problem when uncertainty is captured by a finite set of scenarios. This procedure is then embedded into a sampling mechanism so that realistic instances of the problem can be tackled, namely when the travel times are represented by random vectors with an infinite support. An extensive computational analysis is conducted to assess the methodological developments proposed and the relevance of capturing uncertainty in the problem. Additional insights include the impact of the risk level in the solutions.

Joint optimization of recyclable inventory routing problem under uncertainties in an incentive-based recycling system

Due to the value of resource recovery and the development of a circular economy, waste recycling has gathered global attention. Recently, many emerging cities designed new systems like an incentive-based recycling system (IBRS). In such systems, recyclables are collected through community recycling nodes by offering incentives, then transported to street recycling stations and sorted before being finally recycled. The increased recycling nodes and the incentives enhance the convenience and residents’ enthusiasm for waste recycling, but also intensify the uncertainty of recycling quantities and the complexity of the recycling operation management. Poor recycling operation management may result in increased recycling costs or greater loss of recyclables, which discourages residents from participating in recycling. Based on an existing IBRS, this study investigates the joint optimization problem of the recyclable inventory management at each community recycling node and the vehicle routing from the recycling nodes to the recycling station. A two-stage dual-objective multi-period stochastic programming model is established to minimize the loss of recyclables and logistics costs, which is further reformulated using the weighting method and transportation cost approximation parameters. To solve the reformulated model, a three-phase iterative algorithm is designed by combining the progressive hedging algorithm and route splitting algorithm based on the Lin-Kernighan heuristic. A case study is conducted using data from Shanghai’s IBRS. The proposed joint decision model is superior to separate decisions and the three-phase iterative algorithm can reduce the average total cost by up to 42.12% compared to the genetic algorithm and the Iteration-Move-Search method in the literature. Additionally, a sensitivity analysis is conducted to provide managerial insights.

Emergency medical supply planning considering prepositioning and dynamic in-kind donation management in healthcare coalitions.

This study tackles an integrated emergency medical supply planning problem, which incorporates supply prepositioning and dynamic in-kind donation management in healthcare coalitions. Although this problem is vital for field practice, it is not investigated in the existing emergency supply planning literature. To fill the gap, we propose a two-stage stochastic programming model, which facilitates the planning of emergency medical supply prepositioning before disasters and dynamic supply transshipment and in-kind donation solicitation and distribution after disasters. With a case study on the healthcare coalition of West China Hospital in Sichuan Province of China under the background of the COVID-19 epidemic, the proposed model and seven comparison models are optimally solved to show the effectiveness and benefits of our model. We conduct sensitivity analysis to generate managerial insights and policy suggestions for better emergency medical supply management practices in healthcarecoalitions.

Optimal Placement and Sizing of Distributed Generation in Electrical DC Distribution Networks Using a Stochastic Mixed-Integer LP Model

AbstractThis paper presents a stochastic mixed-integer linear mathematical model for finding the optimal placement and sizing of distributed generation in a DC distribution network, considering the uncertainty of electrical demand and distributed renewable sources. The proposed model accurately represents the original mixed-integer nonlinear model, obtaining a globally optimal solution in less computational time with low errors. The mathematical model allows for considering constraints related to the maximum limits for the penetration of distributed generation, such as those specified by Resolution CREG 174 of 2021. Furthermore, the uncertainties of the electrical demand, wind energy-based distributed generation (DG), and solar energy-based DG are considered in the mathematical models using a two-stage stochastic programming approach. The accuracy and efficiency of the proposed model were tested and validated on a 21-node DC test system from the specialized literature, and the effectiveness and robustness were assessed on a 69-node DC test system. The obtained results show that the proposed stochastic mixed-integer linear mathematical model performs well.

Optimal scheduling of battery energy storage system operations under load uncertainty

This paper investigates the optimal scheduling of battery energy storage system operations considering energy load uncertainty. We develop a novel two-stage distributionally robust optimization model to determine an optimal battery usage schedule that minimizes the worst-case energy costs considering peak load costs. The model leverages deep-learning-based probabilistic forecasting in the construction of the ambiguity set. Specifically, we develop a Deep Autoregressive Recurrent Networks model to generate a probabilistic forecast of energy loads over a time horizon. The output of the forecasting model is then used to construct a marginal-moment ambiguity set for the distributionally robust optimization model. To solve the proposed model, we establish a closed-form characterization of the optimal second-stage objective function value. Leveraging this closed-form expression and using second-order conic duality, we derive an exact single-level mixed integer second-order conic reformulation of the problem. Extensive computational experiments, conducted on a real dataset, demonstrate the value of our proposed model and the effectiveness of the resulting battery schedule. The results show that the proposed model outperforms several benchmarks, including two-stage stochastic programming. Furthermore, the accuracy of the load forecast significantly impacts the effectiveness of the optimal battery schedule in eliminating peak loads by achieving up to 18% reduction in the maximum energy load.

Electrification of transportation: A hybrid Benders/SDDP algorithm for optimal charging station trading

This paper examines the electrification of transportation as a response to environmental challenges caused by fossil fuels, exploring the potential of battery electric vehicles and hydrogen fuel cell vehicles as alternative solutions. However, a significant barrier to their widespread adoption is the limited availability of charging infrastructure. Therefore, this study proposes the development of comprehensive charging stations capable of accommodating both battery and hydrogen vehicles to address this challenge. The energy is purchased from the day-ahead and intraday auction-based electricity markets, where the electricity price is subject to uncertainty. Therefore, a two-stage stochastic programming model is formulated while the price scenarios are generated utilizing a k-means clustering algorithm. Given the complexity of the proposed model, an efficient solution approach is developed through the hybridization of the Benders decomposition algorithm and stochastic dual dynamic programming. In the Benders master problem, day-ahead bidding variables are determined, whereas the Benders sub-problem addresses intraday bidding and charging station scheduling variables, employing stochastic dual dynamic programming to tackle its intractability. Additionally, we transform the mixed integer linear program model of the second stage problem into a linear program, confirming its validity through KKT conditions. Our model provides practical insights for making informed decisions in electricity markets based on sequential auctions. While the bidding curves submitted to the day-ahead market remain unaffected by scenarios, those submitted to the intra-day market show dependence on fluctuations in day-ahead market prices.

Optimal operation of multi-energy carriers considering energy hubs in unbalanced distribution networks under uncertainty

This article presents a two-stage stochastic programming model to address the dispatching scheduling problem in an energy hub, considering an unbalanced active low-voltage (LV) network. A three-phase version of the second-order cone relaxation of DistFlow AC optimal power flow (AC-OPF) is employed to incorporate unbalanced network constraints, while the objective minimizes the Local Energy Community (LEC) operational cost. The model results have been validated using OpenDSS, encompassing energy losses, voltage levels, and active/reactive power. Likewise, a comparative analysis between the three-phase model and the traditional single-phase model, using a modified version of the IEEE European LV Test Feeder as a case study, reveals interesting differences, such that the single-phase model underestimates voltage limits during photovoltaic (PV) system operation and overestimates energy purchased from the main grid, compared with the three-phase model. Similarly, the comparison results reveal that discrepancies between the single and three-phase models intensify as the power injected from PV systems rises. This notably impacts the total energy purchased from the grid, battery operation, and the satisfaction of thermal consumption through electricity. Finally, while the three-phase model offers valuable insights into security levels for voltage and grid energy purchase, its longer computational time makes it more suitable for strategic use rather than daily operational frameworks.

Multistage stochastic programming with a random number of stages: Applications in hurricane disaster relief logistics planning

We consider a logistics planning problem of prepositioning relief commodities in preparation for an impending hurricane landfall. We model the problem as a multi-period network flow problem where the objective is to minimize the total expected logistics cost of operating the network to meet the demand for relief commodities. We assume that the hurricane’s attributes evolve over time according to a Markov chain model, and the demand quantity at each demand point is calculated based on the hurricane’s attributes (intensity and location) at the terminal stage, which corresponds to the hurricane’s landfall. We introduce a fully adaptive multi-stage stochastic programming (MSP) model that allows the decision-maker to adapt their logistics decisions over time according to the evolution of the hurricane’s attributes. In addition, we develop a novel extension of the standard MSP model to address the challenge of having a random number of stages in the planning horizon due to the uncertain landfall time of the hurricane. We benchmark the performance of the adaptive decision policy given by the MSP models with alternative decision policies, including a static policy, a rolling-horizon policy, a wait-and-see policy, and a decision-tree-based policy, all based on two-stage stochastic programming models. Our numerical results and sensitivity analyses provide key insights into the value of MSP in the hurricane disaster relief logistics planning problem.

A two-stage stochastic programming model for comprehensive risk response action selection: A case study in Industry 4.0

Effective project risk management is critical in environments where both micro-level and macro-level risks are present. Traditional models often focus on micro-level risks, neglecting broader macroeconomic uncertainties such as geopolitical instability and supply chain disruptions. This research introduces a two-stage stochastic programming model designed to optimize the selection of Risk Response Actions (RRAs) under uncertainty while addressing both types of risk. The model incorporates “here-and-now” decisions at the planning stage and “wait-and-see” decisions as uncertainties unfold, enabling adaptive risk management throughout the project lifecycle. To solve the model efficiently, we employ an evolutionary algorithm combined with Sample Average Approximation (SAA) to handle the computational complexity of multiple scenarios. The model is applied to a real-world case study involving the integration of IoT and ERP systems in a smart factory in Iran, a project characterized by significant macroeconomic and geopolitical risks. Our key contribution lies in providing a comprehensive risk response strategy selection model that simultaneously addresses micro- and macro-level risks while incorporating strategic flexibility through outsourcing decisions. The results demonstrate that our model outperforms traditional deterministic models, offering enhanced resilience against macro-level risks and improved project performance under uncertainty. These findings provide valuable insights for project managers aiming to increase resilience and adaptability in volatile environments. By integrating both internal and external risk factors, our model offers a robust tool for managing complex projects, enhancing decision-making and project outcomes in uncertain conditions

Empirical risk minimization for big data driven prescriptive analytics: An exploration of two-stage stochastic programs with recourse

In the operations research literature, data driven analyses using big data are receiving more and more interest and attention. However, big data driven operational analyses are still limited in scope, with only unconstrained big data driven problems, e.g. the newsvendor problem, being comprehensively explored. If big data driven analyses are to become a core part of operations research and practice, one must be able to formulate and solve constrained big data driven models. We therefore propose a direct empirical risk minimization (DERM) method for formulating and solving a class of constrained big data driven operation research problems. In big data driven problems, relevant operational prescriptions are dependent on a set of observed features. In this paper, we assume linear feature relationships, which enable us to fundamentally base the DERM method on linear stochastic programming. The assumption of linear feature relationships is also seen in the literature on big data driven newsvendor models. We prove that if a big data driven solution exists, the DERM method will always, for the class of constrained problems we study, arrive at a feasible solution of the initial operational problem. Moreover, we exemplify the DERM method by formulating and solving two specific big data driven operational problems. In the numerical study of the two operational problems, we show, that under linear demand, the DERM method outperforms, with regard to cost, two benchmark methods and a big data driven method from literature.

Optimizing Fleet Size in Point-to-Point Shared Demand Responsive Transportation Service: A Network Decomposition Approach

With increasing urbanization and the demand for efficient, flexible transportation solutions, demand-responsive transportation services (DTRS) has emerged as a viable alternative to traditional public transit. However, determining the optimal fleet size to balance the investment and operational revenue remains a significant challenge for service providers. In this article, we address the optimization of fleet size in point-to-point shared demand DRTS, which widely operates within many cities. To capture the uncertain passenger demands in the future when planning the fleet size currently, we model this problem with a framework of two-stage stochastic programming with recourse. Fleet sizing decisions are made in the first stage before the uncertain demands are revealed. After the uncertainty is revealed, the second stage involves making additional decisions to maximize operational revenue. The objective is to optimize the total revenue of the first-stage decisions and the expected revenue of the recourse actions. To solve this practical problem, we resort to the Model Predictive Control method (MPC) and propose a network decomposition approach that first converts the transportation network to a nodal tree structure and then develops a Nodal Tree Recourse with Dependent Arc Capacities (NTRDAC) algorithm to obtain the exact value of the expected recourse functions. In the experiments, NTRDAC is able to produce results within seconds for transportation networks with over 30 nodes. In contrast, a commercial solver is only capable of solving networks with up to five nodes. The stability tests show that NTRDAC remains robust as the problem size varies. Lastly, the value of the stochastic solution (VSS) was evaluated, and the results indicate that it consistently outperforms the expected value solutions. Numerical experiments show that the performance of the NTRDAC algorithm is quite encouraging and fit for large-scale practical problems.

Integrated emergency medical supply planning considering multiple supply channels in healthcare coalitions

Emergency medical supplies are crucial for the successful disaster response of healthcare coalitions. In practice, a healthcare coalition can obtain emergency medical supplies from three channels, i.e., supply pre-positioning, contracted reserve and emergency procurement, which, however, are not planned integrally in the literature. To fill the research gap, this study proposes a two-stage stochastic programming model for integrated emergency medical supply planning considering multiple supply channels in healthcare coalitions. In the first stage before disasters, decisions on emergency medical supply pre-positioning and signing of two types of medical supply procurement contracts are determined. In the second stage after disasters, decisions of emergency supply procurement, contract implementation, and supply transshipment are optimized based on the first-stage decisions and the realized uncertain disaster impacts. To show the effectiveness of our model and obtain managerial insight, we develop four comparison models and conduct a case study on the healthcare coalition of West China Hospital in China. This study highlights the great benefits of supplementing the pre-positioning of emergency medical supplies with multi-type contracted reserve in healthcare coalitions and emphasizes the importance of strengthening cooperation with suppliers and encouraging all member hospitals to implement the contracted reserve.

Solving a class of two-stage stochastic nonlinear integer programs using value functions

Solving a class of two-stage stochastic nonlinear integer programs using value functions

Logistics planning for direct temporary disaster housing assistance under demand uncertainty

In this paper, we propose and study a framework for disaster housing logistics planning under demand uncertainty. Specifically, we utilize a two-stage chance-constrained stochastic programming model to achieve the balance between logistics operational cost and demand fulfillment especially towards extreme disaster scenarios. To do so, we incorporate two operational modalities, one for the ordinary modality and the other for the emergency modality, and the emergency modality is only allowed to be activated for a certain percentage of scenarios that is specified by the decision maker among all scenarios. The set of scenarios is generated according to a spatial regression model for characterizing the disaster housing demand based on a selected number of independent variables related to both the hazard and socioeconomic factors, which is trained offline from historical data. We conduct a numerical experiment based on Hurricane Ian, and our numerical results show the effectiveness of the proposed approach compared to some standard benchmark approaches. We also highlight the managerial insights for disaster housing logistics planning gained through this numerical experiment.

A low voltage load balancing distribution method considering street information and V2G technology application

The low-voltage distribution network (LVDN) is the final stage in delivering electric energy from power plants to consumers, and its operational condition greatly impacts many power users. While medium-voltage and high-voltage distribution networks can be managed through intelligent digital systems, load imbalance issues in LVDNs often rely on planners’ experience, leading to significant limitations. With advancements in electric vehicle (EV) charging technology and vehicle-to-grid (V2G) technology, where EVs act as distributed energy storage units, bidirectional energy exchange between vehicles and the grid can now contribute to LVDN operation. This paper proposes a low-voltage load distribution planning method that integrates street information and V2G technology. A two-stage stochastic programming mixed-integer model is developed to tackle load imbalance in LVDNs, with the planning scheme derived from solving this model. A case study is presented to verify the effectiveness of the method, demonstrating that incorporating V2G technology enhances load distribution accuracy and reduces reliance on manual planning, improving network stability and operational efficiency.

Adaptive Two-Stage Stochastic Programming with an Analysis on Capacity Expansion Planning Problem

Adaptive Two-Stage Stochastic Programming with an Analysis on Capacity Expansion Planning Problem

Home-based care and center-based care: From being alternatives to being synergistic. Optimization models to support flexible care delivery

The delivery of health care services, especially for patients with chronic conditions and/or requiring cyclical treatment, can be accomplished by resorting to two organizational models: center-based care and home-based care. In developed countries, government pressure to reduce healthcare spending and the COVID-19 pandemic have led to the spread of home-based care. This type of care is proven to outperform the center-based one in terms of cost-effectiveness, patient satisfaction, and adherence to treatment guidelines. However, even for treatments usually suitable to be delivered at home, the patient's health status or other constraints may make it inappropriate to deliver service at the patient's home. This calls for a new organizational model, referred to as flexible care, where home-based and center-based care are not seen as mutually exclusive models but as options that can be activated according to the patient's and provider's needs. This paper presents a novel network-based deterministic optimization model and two matheuristics to address a scheduling problem typically faced by providers adopting a flexible care model. The model considers a provider relying on a treatment room with a fixed number of medical chairs, a fleet of vehicles, and a team of operators. It allows for determining on which days of the planning horizon, in which setting (home or center), and by which operator each patient will be treated. The model takes into account patients’ preferences and considers two objective functions: minimizing provider costs and patients’ travel time. In addition, we propose a two-stage mixed-integer stochastic programming model with recourse actions. This model allows incorporating the uncertainty due to the occurrence of adverse events. Adverse events are sudden changes in the patient's condition randomly happening at a specific point in the planning horizon. These events render the patient unsuitable for home care and require them to be visited at the center from that moment onward. The models have been inspired by a real case and tested on multiple random instances.

A two-stage stochastic programming framework for blood product inventory management with ABO substitution and lateral transshipment

This work proposes a two-stage stochastic optimization model for managing the inventory of a two-echelon blood supply chain comprised of a blood center and a network of hospitals and transfusion points. The proposed model identifies optimal distribution quantities across the supply chain, considering demand stochasticity and product perishability. Furthermore, the model utilizes redistribution policies, such as lateral transshipment and product substitution based on ABO compatibility, as corrective actions to minimize outdates (i.e., expired product) and shortages across the network. Using a computational experiment that includes a network with one regional blood center and twenty hospitals, we show that inventory management policies that include both ABO substitution and transshipment can significantly decrease the expected total network costs. Similarly, we see corresponding reductions in the expected number of outdated units and emergency requests to external networks for additional supply to satisfy unmet demand.

Fleet repositioning in the tramp ship routing and scheduling problem with bunker optimization: A matheuristic solution approach

This paper investigates an important planning problem faced by dry bulk shipping operators, referred to as the Tramp Ship Routing and Scheduling Problem with Bunker Optimization (TSRSPBO). The problem is to maximize the overall profit of a fleet of vessels by selecting cargoes and determining ship routes and schedules. We consider this problem under a set of practically relevant features such as flexibility in cargo quantities, as well as bunkering decisions on where to procure fuel and how much. As a particularly novel feature, we address the regional allocation of vessels at the end of the planning period to be well prepared for meeting (uncertain) future demand. To incorporate this, we consider the TSRSPBO as a two-stage stochastic programming problem, where cargo selection, routing, and bunkering decisions are solved in the first-stage problem, and the recourse cost of fleet repositioning is considered in the second stage. We present arc flow and path flow formulations, where the latter employs a priori generation of feasible routes as input. For solving realistically sized instances, we propose a matheuristic based on an Adaptive Large Neighborhood Search (ALNS) framework that iteratively generates columns and solves the path flow model. Computational experiments based on real data show that this matheuristic finds high-quality solutions for large test instances with 120 cargoes, 30 vessels, and ten bunker ports in less than one hour. Also, considering the TSRSPBO as a two-stage stochastic problem achieves the highest profits and is solved almost as quickly as the deterministic problem variant.