Two-stage Stochastic Programming Research Articles

Technology and energy choices for fleet asset decarbonization under cost uncertainty

The road freight transportation sector faces significant challenges in reducing greenhouse gas emissions due to uncertainty in the costs and trajectories of low-emission technologies. This work introduces the Fleet Assets Decarbonization Problem (FADP) to guide and facilitate fleet owners’ investment decisions by providing cost-minimizing decarbonization pathways for ground fleets under decarbonization targets. It integrates strategic vehicle and infrastructure composition decisions with tactical fuel and energy usage decisions. We propose a stochastic programming formulation that incorporates cost uncertainty and optimizes the fleet decarbonization strategy over a range of scenarios. We provide a tailored rolling-horizon two-stage stochastic programming approximation solution approach and apply our formulation and solution approach to the heavy-duty truck fleet of a large U.S. consumer packaged goods company.

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.

A Lagrangian relaxation approach for algae-based biofuel supply chain network design under uncertainty and pricing issue.

Biofuel has gained significant attention as a potential source to meet fuel demands instead of fossil fuel. The price of biofuel and alternative fuel have a considerable impact on biofuel demand. Thus, it is important to design a biofuel supply chain network that incorporates the biofuel price into an elastic demand. More precisely, a variable demand, including customer importance level (to the environment), biofuel price, and substituted fuel price, is considered in this research. Furthermore, this research presents a bi-objective mixed-integer quadratic formulation that aims to maximize the total profit of the supply chain and carbon absorption in harvesting areas. The problem is solved by the ε-constraint and lagrangian relaxation methods due to its complexity. Moreover, substituted fuel price uncertainty is addressed by two-stage stochastic programming. Finally, a real case study utilizing the data envelopment analysis approach is applied to assess the efficiency and currency of the addressed model. Several consequences are illustrated in the case study, such as rich areas for exporting algae, suggesting hub locations for biofuel production, etc.

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

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

Problem definition: Multistage stochastic programming is a well-established framework for sequential decision making under uncertainty by seeking policies that can be dynamically adjusted as uncertainty is realized. Often, for example, because of contractual constraints, such flexible policies are not desirable, and the decision maker may need to commit to a set of actions for a certain number of periods. Two-stage stochastic programming might be better suited to such settings, where first-stage decisions do not adapt to the uncertainty realized. In this paper, we propose a novel alternative approach, named as adaptive two-stage stochastic programming, where each component of the decision policy requiring limited flexibility has its own revision point, a period prior to which the decisions are determined at the beginning of the planning until this revision point, and after which they are revised for adjusting to the uncertainty realized thus far until the end of the planning. We then analyze this approach over the capacity expansion planning problem, that may require limited flexibility over expansion decisions. Methodology/results: We provide a generic mixed-integer programming formulation for the adaptive two-stage stochastic programming problem with finite support, in particular, for scenario trees, and show that this problem is NP-hard in general. Next, we focus on the capacity expansion planning problem and derive bounds on the value of adaptive two-stage programming in comparison with the two-stage and multistage approaches in terms of revision points. We propose several heuristic solution algorithms based on this bound analysis. These algorithms either provide approximation guarantees or computational advantages in solving the resulting adaptive two-stage stochastic problem. Managerial implications: We provide insights on the choice of the revision times based on our analytical analysis. We further present an extensive computational study on a generation capacity expansion planning problem with different generation resources including renewable energy. We demonstrate the value of adopting adaptive two-stage approach against the existing policies under limited flexibility and highlight the efficiency of the proposed heuristics along with practical implications on the studied problem. Funding: This work was supported by the National Science Foundation [Grant 1633196] and the Office of Naval Research [Grant N00014-18-1-2075]. Supplemental Material: The online appendices are available at https://doi.org/10.1287/msom.2023.0157 .

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.

Optimal scheduling of smart home appliances with a stochastic power outage: A two-stage stochastic programming approach

This study primarily concentrates on enhancing the scheduling of electric appliances within a smart home equipped with a photovoltaic solar array and a storage battery capable of redistributing excess electricity to the grid. Furthermore, the investigation takes into consideration the unpredictability of a power outage, analyzing the scheduling of these appliances to minimize consumer electricity expenses. To tackle this issue, a two-stage stochastic programming methodology is developed to effectively model the uncertainty surrounding both the onset time and duration of power outages. Additionally, a sample average approximation algorithm (SAA) is devised to efficiently address the problem. Through extensive computational experiments, the outcomes demonstrate that the SAA yields shorter CPU processing times, albeit without guaranteeing optimal solutions. The total average percent deviation of the SAA's upper bound from its lower bound and the optimal solution stands at 4.16 % and 4.5 %, respectively. Moreover, it is demonstrated that utilizing the stochastic approach, as opposed to the deterministic one, can enhance solution quality by 6.2 %. Furthermore, a comprehensive sensitivity analysis is provided, focusing on probability distribution functions of outage start time and duration, alongside an analysis of the solar panel and battery storage capacities. It is revealed that when adopting a Normal distribution instead of a Uniform distribution, the performance of the SAA experiences a slight decline.

Resilient transportation network design with disruption uncertainty and lead times

Cost-efficient and reliable transports are needed to supply products competitively. Thus, particularly in increasingly complex and global supply chains, identifying the optimal transportation mode is a critical decision. Transportation modes, however, are prone to disruptions, such as hurricanes, low water levels, or port shutdowns, resulting in transportation stops and cost increases. To counteract these disruptions, different resilience strategies are studied to increase the capability of a network to withstand, adapt, and recover from disruptions. For a cost-optimal use, it is necessary to determine the optimal mix of strategic, tactical, and operational strategies.We provide a decision-support model that decides on the optimal mix of resilience strategies, such as multi-sourcing, inventory, or operational re-routing, for a supply chain with transportation disruption uncertainty to minimize total expected costs. The problem is formulated as a two-stage stochastic mixed-integer linear program that explicitly considers lead times. To handle large instances, we propose a Benders decomposition approach enhanced through lower-bound lifting and valid inequalities, branch-and-benders-cut, and a warm-start heuristic. Computational experiments show that large instances can be solved to near-optimality, whereas a commercial solver does not find feasible solutions.We present a case study for a company’s inbound supply chain design with recurring transportation cost uncertainty. Considering disruption and lead time effects, a mix of resilience strategies from strategic to operational level leads to cost improvements of up to 50%. Furthermore, we show that the ability to predict disruptions can further reduce resilience-related costs by 10% if sufficient operational re-routing capacities are available.

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.

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.

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.

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.

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.

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.

Optimal planning and management of the energy–water–carbon nexus in hybrid AC/DC microgrids for sustainable development of remote communities

This paper presents a two-stage stochastic mixed-integer nonlinear programming model, which is linearized to a mixed-integer linear programming (MILP) form, for optimizing the energy–water–carbon (EWC) nexus in remote AC/DC microgrids aimed at sustainable community development. The model optimizes the selection, location, and operation of diesel generators, voltage source converters, photovoltaic systems, wind turbines, and battery energy storage systems while managing CO2 emissions. It accurately models the AC/DC microgrid’s operation, determining optimal voltages, currents, and converter states (inverter or rectifier). Uncertainties in power demand, water consumption, solar irradiation, and wind speed are modeled using stochastic scenarios generated via k-means clustering. The MILP model is implemented in GAMS and solved with the commercial solver CPLEX, ensuring global optimization. Results show that integrating distributed energy resources, compared to a case without these elements, reduces CO2 emissions of the diesel generators and the operational costs of the EWC nexus, highlighting the proposed approach’s environmental and economic benefits. These findings underline the importance of incorporating distributed energy resources in the sustainable development of microgrids.

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.