Two-stage Stochastic Optimization Model Research Articles

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 Similarly, we see corresponding reductions in the expected number of outdated units and emergency requests to external networks for additional supply to satisfy unmet demand.

Multi-objective optimization for a green forward-reverse meat supply chain network design under uncertainty: Utilizing waste and by-products

The essential requirement for food stands as a pivotal human need, exerting significant ecological impact from production to consumption. Meat, a key dietary component, offers essential nutrients vital for human health. This paper presents a bi-objective two-stage stochastic optimization model for a green forward-reverse meat supply chain network design, addressing both economic and environmental concerns throughout the chain. In the forward flow, the supply chain manages various meat products consist of fresh, processed, and frozen meat products, ensuring their eco-efficient production and distribution. Meanwhile, in the reverse flow, waste and by-products generated during the production process are repurposed and reused. The study promotes environmental sustainability by repurposing waste, utilizing by-products, and minimizing carbon emissions. The proposed model is solved using popular exact ε-constraint method for smaller instances and Non-dominated Sorting Genetic Algorithm II (NSGA-II), Multi-Objective Particle Swarm Optimization (MOPSO), and Strength Pareto Evolutionary Algorithm 2 (SPEA2) meta-heuristic algorithms are employed for larger instances. SPEA2 outperforms both MOPSO and NASG-II, demonstrating less average gap that is 0.38% and 0.76%, respectively. Additionally, the findings reveals that an average 2.5% reduction in environmental impacts associated with an average 5% decrease in profit. The noteworthy outcomes of the research empower managers to navigate the implications of fluctuating demand effectively. Moreover, it is crucial to underscore the effects of conversion rates, particularly those associated with the manufacturing process. An excessively high conversion rate can negatively impact profitability and worsen environmental issues. Conversely, lowering the conversion rate can enhance profitability and mitigate environmental impacts.

Impact of power outages depends on who loses it: Equity-informed grid resilience planning via stochastic optimization

This research presents a novel approach for enhancing power grid resilience with a focus on social equity in light of increasing natural disasters. Utilizing a two-stage stochastic optimization model for flood mitigation investments, we optimize substation hardening and power flow decisions to minimize the weighted combination of expected load shed and expected equity metric. Our method uniquely approximates community power outages stemming from transmission grid disruptions, sidestepping the complexities of the distribution grid, and it incorporates two equity metrics (affected population and duration of loss) to address the uneven postdisaster well-being loss across communities. Our approach’s novelty lies in integrating this equity-informed resilience model with realistic flood scenario generation and utilizing a large-scale synthetic but a realistic power grid of Texas. The findings highlight the importance of the composite objective function in altering power flow decisions to prioritize electricity provision and save communities in disadvantaged areas even without investing in substation hardening. The results also quantify the equity and load shed benefits of substation hardening as a function of the investment budget with a parameterized analysis. With an attention to equity, power outages increase in nonvulnerable communities — a trade-off made to mitigate well-being loss in the most vulnerable areas. We further explore a justice model inspired by the government’s Justice40 initiative but find it less effective than our equity-informed models at preventing well-being loss. Our research, enriched by a comprehensive sensitivity analysis, offers valuable insights for policymakers, grid operators, and utilities aiming for a more resilient and equitable power grid.

Managing oversaturation in BRT corridors: A new approach of timetabling for resilience enhancement using a tailored integer L-shaped algorithm

Bus rapid transit (BRT) is a high-capacity public transport system that typically operates along urban transit corridors with dense travel demand. Maintaining the efficiency and stability of the BRT is paramount for daily transport operations. Owing to the difficulty in ensuring an exclusive right-of-way along the entire route, stochastic congestion events may occur resulting from road segments without dedicated BRT lanes. This may lead to volatility in travel time and resulting in passenger stranding, determined as common-case disruptions in this study. These common-case disruptions frequently occur in the daily operation of oversaturated BRT routes. To manage and mitigate their negative impacts, a novel timetabling problem for enhancing the resilience of a BRT system was proposed to assess the ability of the system to withstand and recover from these disruptions. We formulated the problem as a two-stage stochastic mixed-integer optimization model and designed an exact algorithm based on a tailored integer L-shaped method. We then analyzed the structural properties of our model and developed several acceleration techniques to further improve the efficiency of the algorithm. The computational results show that the proposed algorithm outperforms the commercial solver in large-scale instances and can provide near-optimal solutions when the commercial solver is invalid. Besides, external comparisons with dynamic programming also demonstrate the superiority of the proposed algorithm in solution efficiency. Compared with the benchmark timetabling problem, which aims to reduce passenger waiting time, the proposed method can efficiently reduce the duration of the oversaturation period by 35.3%.

Comparing resilience strategies for a multistage green supply chain to mitigate disruptions: A two-stage stochastic optimization model

Supply chain disruptions, notably exemplified by the COVID-19 pandemic, have substantial impacts, requiring effective resilience strategies. This study introduces a comprehensive multi-stage, multi-period green supply chain design model, incorporating six resilience strategies to tackle downstream and upstream disruptions. A rigorous two-stage stochastic optimization approach is employed to manage parameter uncertainties and disruptions. The objectives include minimizing disruption costs and CO2 emissions, for a chain that operates under cap-and-trade emission regulation. The findings, derived from a numerical experiment and sensitivity analysis, offer the optimal supply chain structure and effective resilience strategies to mitigate disruptions. The demand structure for essential and non-essential products during disruptions are explored, emphasizing proactive resource allocation. In particular, the impact of facility capacity reductions underscores the importance of capacity management. Sensitivity analyses reflect the trade-offs involved in capacity management strategies, emphasizing the critical need to maintain an optimal level of capacity to prevent service level degradation. Additionally, examination of carbon abatement regulations reveals the intricate balance between environmental responsibility and economic efficiency. In summary, this study addresses supply chain disruptions and offers actionable insights for supply chain managers and policy makers. The research highlights the importance of specific and data-driven resilience strategies to optimize supply chain performance in face of disruptions.

Residential PV-battery scheduling with stochastic optimization and neural network-driven scenario generation

Residential end-users have increasingly been attracted to installing behind-the-meter (BTM) photovoltaic (PV) and battery energy storage systems (BESS). The inherent stochastic nature of PV introduces challenges to energy management for PV-BESS owners. This work presents a two-stage stochastic optimization (SO) framework for battery operation that takes into account uncertainties in PV outputs to minimize daily grid consumption and battery degradation. A combination of a feed-forward neural network (FFNN) and an error analysis statistical technique is proposed to generate a highly accurate scenario set for estimating PV behavior. Then, the scenario set is reduced using a backward scenario reduction technique to address the issue of dimensionality that SO suffers from. The reduced scenario set is then fed into the two-stage SO model to calculate expected electricity costs and battery degradation for the day-ahead PV-BESS problem. The suggested model's performance in controlling BTM batteries is evaluated using real-world data collected from smart meters installed in a house in Aran Islands, Ireland. The results show that the model can effectively withstand sudden changes in the PV output and generate results that are comparable to an ideal case with accurate PV data available.

A Revisit to Sunk Cost Fallacy for Two-Stage Stochastic Binary Decision Making

This paper undertakes a revisit of the sunk cost fallacy, which refers to the tendency of people to persist investing resources into something, even if it is destined to have no good outcome. We emphasize that the utilities associated with different alternatives are not static for decision makers, which is exactly opposite to the traditional perspective. This paper argues that the utility of an option may change due to the choice of another option, suggesting that decisions considered irrational by the traditional analytical method, i.e., sunk cost fallacy, may be rational. We propose a novel analytical method for decision making with sunk cost when considering the utility change and validate the effectiveness of this method through mathematical modeling and computational experiments. This paper mathematically describes such decision-making problems, analyzing the impact of changes in the utilities across different alternatives on decision making with a real-world example. Furthermore, we develop a two-stage stochastic optimization model for such decision-making problems and employ the sample average approximation (SAA) method to solve them. The results from computational experiments indicate that some decisions traditionally considered irrational are, in fact, rational when the utility of an option changes as a result of choosing another option. This paper, therefore, highlights the significance of incorporating utility changes into the decision-making process and stands as a valuable addition to the literature, offering a refreshed and effective decision-making method for improved decision making.

Optimal Prosumer Operation with Consideration for Bounded Rationality in Peer-to-Peer Energy Trading Systems

With the large-scale development of distributed energy on the demand side, the trend of “supply exceeding demand” has gradually become prominent, and regional peer-to-peer (P2P) energy trading has become an important measure to improve the local consumption of distributed energy. However, most existing studies usually assume that prosumers behave entirely rationally with the goal of maximum benefit, which has been proved to deviate from the observed actual behaviors. Aiming at the optimal energy of prosumers participating in the P2P market, a prospect theory-based two-stage stochastic optimization model considering the bounded rationality was proposed to accurately simulate the decision-making behavior. Then, a benefit maximization model for the energy trading service provider (ETSP) was constructed considering the power flow constraint to ensure the safe operation of the system. Finally, an improved R-ADMM algorithm considering timeout was proposed to solve the above model and improve the convergence speed. The effectiveness of the proposed model and algorithm was verified via simulation.

Optimal Operation and Market Integration of a Hybrid Farm with Green Hydrogen and Energy Storage: A Stochastic Approach Considering Wind and Electricity Price Uncertainties

In recent years, growing interest has emerged in investigating the integration of energy storage and green hydrogen production systems with renewable energy generators. These integrated systems address uncertainties related to renewable resource availability and electricity prices, mitigating profit loss caused by forecasting errors. This paper focuses on the operation of a hybrid farm (HF), combining an alkaline electrolyzer (AEL) and a battery energy storage system (BESS) with a wind turbine to form a comprehensive HF. The HF operates in both hydrogen and day-ahead electricity markets. A linear mathematical model is proposed to optimize energy management, considering electrolyzer operation at partial loads and accounting for degradation costs while maintaining a straightforward formulation for power system optimization. Day-ahead market scheduling and real-time operation are formulated as a progressive mixed-integer linear program (MILP), extended to address uncertainties in wind speed and electricity prices through a two-stage stochastic optimization model. A bootstrap sampling strategy is introduced to enhance the stochastic model’s performance using the same sampled data. Results demonstrate how the strategies outperform traditional Monte Carlo and deterministic approaches in handling uncertainties, increasing profits up to 4% per year. Additionally, a simulation framework has been developed for validating this approach and conducting different case studies.

Virtual Power Plant’s Optimal Scheduling Strategy in Day-Ahead and Balancing Markets Considering Reserve Provision Model of Energy Storage System

In recent years, with the rapid increase in renewable energy sources (RESs), a Virtual Power Plant (VPP) concept has been developed to integrate many small-scale RESs, energy storage systems (ESSs), and customers into a unified agent in the electricity market. Optimal coordination among resources within the VPP will overcome their disadvantages and enable them to participate in both energy and balancing markets. This study considers a VPP as an active agent in reserve provision with an upward reserve capacity contract pre-signed in the balancing capacity (BC) market. Based on the BC contract’s requirements and the forecasted data of RESs and demand, a two-stage stochastic optimization model is presented to determine the VPP’s optimal scheduling in the day-ahead (DA) and balancing energy (BE) markets. The probability of reserve activation in the BE market is considered in this model. The ESS’s reserve provision model is proposed so as not to affect its schedule in the DA market. The proposed optimal scheduling model is applied to a test VPP system; then, the effects of the BC contract and the probability of reserve activation on the VPP’s trading schedule are analyzed. The results show that the proposed model has practical significance.

Day-ahead bidding and dispatch strategy of novel battery charging and swapping station under multiple uncertainties

The novel battery charging and swapping station (NBCSS) has great operational flexibility due to its integration of wind power, photovoltaic power, gas turbine and energy storage. This paper presents a day-ahead bidding and dispatch strategy of NBCSS under multiple uncertainties, which consists of battery demand, market price, renewable energy and load. Firstly, the optimized backup method is used to address the uncertainty of battery demand. Then, a two-stage stochastic robust optimization model is further applied to address the uncertainties of market price, renewable energy and load. The model’s goal is to minimize the total operational cost in the worst-case scenario, which is limited by the corresponding uncertainty set. Especially, to control the conservatism of the model, the total adjustable budget and temporal adjustable budget of uncertainties are considered together. Next, the two-stage model is solved efficiently based on the strong duality theory and column-and-constraint generation (C&CG) method. Finally, case studies are given to verify the effectiveness and performance of the proposed model. The results demonstrate that the NBCSS can be efficiently deployed, enabling market arbitrage and reducing the overall cost. Additionally, the conservatism of the proposed model can be adjusted according to the budgets of uncertainties.

An operational scheduling framework for Electric Vehicle Battery Swapping Station under demand uncertainty

This paper proposes a novel mathematical optimization model aimed at incorporating demand response strategies in the operational scheduling problem of Electric Vehicle (EV) Battery Swapping Stations (BSS). The primary goal of the model is to minimize the operational costs while accounting for intricacies arising from diverse charging modes, variations in battery capacities, charger availability, different states of charges (SOC) of incoming discharged batteries, time-of-use power prices, power constraints and fluctuating customer demand patterns. Furthermore, the model takes into consideration the effect of faster charging modes on battery health to ensure a sustainable operation. The proposed model is further extended to incorporate the effect of swapping demand uncertainty on the operational decision-making of the BSS, and a two-stage stochastic optimization model with a recourse approach is proposed. The proposed stochastic optimization model is tested with relevant case studies, and the results demonstrate the efficacy of the model in providing a risk-neutral solution by determining the optimal demand satisfaction pattern and calculating the number of additional batteries needed to minimize demand shortfalls across all scenarios. The proposed scheduling framework holds the promise of improving both the efficiency of the BSS operations and customer satisfaction, paving the way for accelerated EV adoption.

Multi-Objective Optimization of Car Sharing Points Under Uncertainty for Sustainable Transportation

As an important part of the sharing economy, the usage of car sharing increases world widely with the help of developments in the technology. Especially after COVID-19 the demand for private car ownership and car sharing systems increased tremendously. Therefore, its market share attracts new investors and causes existing service providers to enlarge their service area. In this article, a novel multi-objective location-dependent two-stage stochastic optimization model is proposed to determine the most appropriate locations for car sharing system and allocate the demand to these locations. The model is applied to determine the best locations among 15 candidates, and three objectives are considered, which are the minimization of total cost that comprises locating costs minus income from satisfying the demand, minimization of CO <formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex>$_2$</tex></formula> emission occurs by the usage of car sharing system's cars and minimization of average unsatisfied demand. Both location-independent and location-dependent demands are taken into account. The proposed model delivers a more precise decision process framework for problems include stochasticity and multiobjectivity, and it easily can be implemented to any region, providing region sensitive parameters.

Flexible storage yard management in container terminals under uncertainty

As the port bottleneck shifts from the shore side to the yard side, yard management is crucial to the operation efficiency of the container port. This paper studies the yard management problem in container terminals under the flexible storage strategy which does not fix the size of the storage templates in the yard. The problem considers the uncertainty of container demand and the flexibility of the block and jointly optimizes the storage space allocation, yard crane deployment, and container allocation problems. A two-stage stochastic optimization model under uncertainty is established to minimize the port operation cost. Two heuristic algorithms, a column generation-based heuristic algorithm and a tabu search algorithm, are proposed to solve the model. Numerical experiments validate the effectiveness of the proposed algorithms. Furthermore, some management implications for port operators are proposed according to the results of sensitivity analysis experiments.

Stochastic generation maintenance scheduling with inertia-dependent primary frequency regulation constraints

The problem of insufficient inertia and frequency security becomes a critical concern in generation maintenance scheduling due to the increasing penetration of renewable energy sources (RESs). To address the issue, this paper presents a stochastic generation maintenance scheduling with inertia-dependent primary frequency regulation constraints. First, inertia-dependent primary frequency regulation constraints consisting of the minimum inertia requirement, frequency nadir, and quasi-steady-state frequency deviation are formulated based on the dynamic frequency characteristic of power systems in response to large disturbances. Then, a two-stage stochastic optimization model for generation maintenance scheduling including frequency security constraints is proposed. Finally, a solution methodology based on Benders decomposition (BD) is used to promote the calculation speed of the proposed model. Numerical simulations demonstrate that the proposed generation maintenance scheduling could address the potential frequency security issues and promote integration of RESs associated with calculation speed.

Cooperative mechanisms for multi-energy complementarity in the electricity spot market

Promoting a diversified and sustainable energy mix in the electricity market necessitates the implementation of multi-energy complementarity. However, the absence of effective cooperative mechanisms among diverse power sources causes a significant challenge in maximizing the overall economic benefits of multi-energy complementarity and fostering individual cooperative willingness. This paper employs cooperative game theory to investigate the joint offering and operation of multiple complementary power sources in the electricity spot market, including wind, thermal, and pumped storage. A two-stage stochastic optimization model is established to co-optimize offering strategies at the day-ahead stage and operation policies at the balancing stage for the multi-energy alliance. The Shapley value is used to allocate profits fairly among various power sources. Simulation results show that two- and three-party alliances among wind, thermal, and pumped storage can lead to multiple-win situations in the spot market. The proposed model effectively couples the day-ahead and balancing markets, resulting in higher profits compared to the scenario without coupling. Both wind-thermal and wind-thermal-pumped storage alliances prove highly effective, with a nearly 100% reduction in imbalance power and approximately a 12% increase in wind power absorption. Relatively, the latter alliance performs slightly worse than the former. Furthermore, the wind-thermal-pumped storage alliance achieves a 4.48% carbon emission reduction, surpassing the wind-thermal and wind-pumped storage alliance. This study reveals the cooperation mechanism and its influencing factors among diverse power sources. It provides valuable decision support for stakeholders to achieve effective multi-energy complementarity, mitigate imbalance power, reduce carbon emissions, and increase renewable energy absorption.

Two-stage travel itinerary recommendation optimization model considering stochastic traffic time

In an increasingly competitive market, most online tour operators have launched personalized travel itinerary recommendation services. Different from previous studies, this research considers the travel itinerary planning problem under total time limit and uncertain traffic time. This problem requires two stages of decision-making: firstly, selecting the tourist attractions to visit from the candidate attractions based on maximizing the popularity utility of tourists; secondly, planning the tourist attraction visit sequence under stochastic traffic time to maximize the tourist activity utility. Therefore, a two-stage stochastic optimization model with chance constraint is constructed and then solved by the sample average approximation (SAA) method. In order to verify the effectiveness of our model, we introduced two benchmark models for comparative analysis. The results show that in the worst case, our model increases the reliability level of travel itineraries by almost 40% compared with the two benchmark models. In addition, case studies of two large cities, i.e., Nanjing and Beijing, indicate that a tour operator can optimize travel itinerary recommendations by improving tourists utility as well as their own profitability without much loss of reliability.

Evaluation of network expansion decisions for resilient interdependent critical infrastructures with different topologies

Resilient interdependent critical infrastructures (CIs) can better withstand cascading failures in disruptive events. This study proposes network expansion as a resilience improvement strategy for interdependent CIs and evaluates the influence of topology in interdependent network design for resilience optimization under disruption uncertainty. A resilience score consisting of network complexity and unmet demand metrics is introduced to quantify the resilience of expanded networks. Five synthetic interdependent network instances with random and hub-and-spoke (i.e., cluster) topologies are generated to represent CIs with heterogeneous node functions. Different network expansion opportunities are considered and critical node disruption scenarios are used to evaluate the impact of uncertain disruptions. We apply a two-stage stochastic multi-objective resilience optimization model to determine strategic investment decisions using the expected total cost and expected resilience score as competing objectives. Compromise solutions of expanded network designs are identified from Pareto optimal solutions and they are characterized according to their graph properties. The results show that expanded networks have improved resilience and the extent of improvement is affected by the network topology and type of disruption. Under critical node disruptions, a random network is more resilient than a hub-and-spoke structure due to its better connectivity. Characteristics of highly connected interdependent networks are high average node degree, high clustering coefficient, and low average shortest path length. Resilience improvement is more limited in expanded networks with a hub-and-spoke structure due to the negative impact of hub failures.

Generative adversarial network assisted stochastic photovoltaic system planning considering coordinated multi-timescale volt-var optimization in distribution grids

In this paper, a novel photovoltaic (PV) system planning framework is proposed for distribution grids. The main contribution of this work is that the volt-var control (VVC) capability of PV inverters is duly considered during the planning stage to reduce the expected power loss cost and meanwhile counteract uncertain voltage fluctuation and deviation caused by random PV production and load demand. Specifically, the proposed PV system planning framework is formulated as a two-stage stochastic optimization model, where the first stage is to determine the planning decisions of PV systems before the uncertainty realization, and after the uncertainty is observed, the second stage coordinates PV inverters and existing legacy VVC devices (capacitor banks) to minimize the power loss and voltage fluctuation in a multi-timescale manner. To solve the formulated complex optimization problem, the original model is decoupled and solved using an iterative solution method where commercial solvers can be directly used to obtain the optimal solutions. Besides, a data-driven based scenario selection method based on Generative Adversarial Network (GAN) is proposed to capture the uncertainties of PV production and load demand with the full diversity of behaviours. Finally, the proposed framework is tested on IEEE 37-node and 123-node test systems to demonstrate the effectiveness of the proposed method. This paper can provide important insights into investment strategies of renewable resources designed by power grid authorities, which can benefit from the proposed framework to maintain the reliable system operation and reduce the renewable energy curtailment.

A two-stage stochastic optimization model for port infrastructure planning

A two-stage stochastic optimization model for port infrastructure planning