Two-stage Stochastic Programming Research Articles

An Integrated Predictive Maintenance and Operations Scheduling Framework for Power Systems Under Failure Uncertainty

Maintenance planning plays a key role in power system operations under uncertainty as it helps providers and operators ensure a reliable and secure power grid. This paper studies a short-term condition-based integrated maintenance planning with operations scheduling problem while considering the possible unexpected failure of generators as well as transmission lines. We formulate this problem as a two-stage stochastic mixed-integer program with failure scenarios sampled from the sensor-driven remaining lifetime distributions of the individual system elements whereas a joint chance constraint consisting of Poisson Binomial random variables is introduced to account for failure risks. Because of its intractability, we develop a cutting-plane method to obtain an exact reformulation of the joint chance constraint by proposing a separation subroutine and deriving stronger cuts as part of this procedure. We also derive a second-order cone programming-based safe approximation of this constraint to solve large-scale instances. Furthermore, we propose a decomposition algorithm implemented in parallel fashion for solving the resulting stochastic program, which exploits the features of the integer L-shaped method and the special structure of the maintenance and operations scheduling problem to derive valid and stronger sets of optimality cuts. We further present preprocessing steps over transmission line flow constraints to identify redundancies. To illustrate the computational performance and efficiency of our algorithm compared with more conventional maintenance approaches, we design a computational study focusing on a weekly plan with daily maintenance and hourly operational decisions involving detailed unit commitment subproblems. Our computational results on various IEEE instances demonstrate the computational efficiency of the proposed approach with reliable and cost-effective maintenance and operational schedules. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms—Discrete. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2022.0154 .

A Three-Stage Stochastic Model to Improve Resilience with Lateral Transshipment in Multi-Period Emergency Logistics

Driven by the growing threat of natural disasters caused by climate change, there is an urgent need to strengthen the emergency rescue logistics network. However, insufficient research has been conducted on optimizing both pre-disaster preparation and post-disaster response, resulting in lower resilience and inefficiency of emergency logistics management. To this end, this study explores the optimization of emergency rescue resource allocation and transportation network design, considering the uncertainty and multi-period nature of natural disaster rescue. By employing a lateral transshipment strategy, a three-stage stochastic programming model is established, which aims to balance economic benefits with the need for devastations, thereby enhancing the resilience of the logistics network. Numerical experiments verify the effectiveness of the proposed model with different instances and the performance of the lateral transshipment strategy by comparing it with a two-stage stochastic programming model. Sensitivity analysis is performed on the costs of constructing a depot and the penalties for unmet needs. The analysis yielded valuable insights that can be used to enhance emergency rescue operations, supply chain network design, and logistics network design. The research outcome can benefit emergency responders and logistics professionals in optimizing their operations.

Sustainable supply chain planning with the flexible provisions of inter-period emission credits borrowing and banking under a multi-period carbon trading scheme

Although the carbon emission trading regulation proves its advantage of reducing abatement costs, the timing of operating activities makes overage or shortage of allocated emission allowances quickly happen in single-year compliance, thus increasing abatement costs. The provision of inter-period emission credits banking and borrowing in a multi-period compliance phase can give emitters flexibility in emission compliance, which practices world-widely. In this study, we investigate the impact of carbon trading with inter-temporal emission credits borrowing and banking on a production–inventory–distribution system’s supply chain network decision-making and its compliance strategies by proposing a novel two-stage stochastic programming. A scenario-based robust optimization approach is presented with a post-optimization stage to obtain robustness-profit-environment balanced decision-making considering uncertain demand and stochastic carbon-trading price. The results suggests emitters should adopt specific greener techniques to reduce related emissions with priority under carbon-trading policies. The results also show the provision of carbon credits borrowing and banking has a positive influence on enhancing the efficiency in supply chain economic and ecological performance. Besides, given the flexible provisions, the manager can reallocate emission credits according to its business’s operation, and transfer more emission credits to periods of higher carbon-trading price. This findings also suggest banking in early periods instead of carelessly borrowing from future, for uncertainty in future business might make it harder or impossible to comply with emission regulation at last. To regulators, it should prevent borrowing explicitly or allow it only to a limited extent.

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.

Integrated multi-product reverse supply chain design and disassembly line balancing under uncertainty

End-of-life (EOL) product recycling has received increasing attention because of potential environmental, social and economic benefits. A well-designed reverse supply chain (RSC) can efficiently handle EOL products. As the critical activity in the RSC, the disassembly process decomposes collected EOL products into components to fulfill the demands of remanufacturing plants. Efficiently coordinating RSC design and disassembly line balancing decisions may improve the whole system performance significantly, especially when facing multiple EOL products and uncertainty. This paper investigates a novel integrated RSC design and disassembly line balancing problem to handle multiple EOL products where the supply of EOL products, the demand for components, and task times in disassembly are stochastic. This complex problem needs to jointly determine the number and locations of disassembly plants, disassembly equipment procurement, disassembly line balancing, and inventory levels of both EOL products collected and components dismantled. The objectives are to maximize the expected profit and minimize the carbon emissions simultaneously. For the problem, a bi-objective two-stage stochastic programming model is formulated and an exact ɛ-constrained method is proposed, transforming the bi-objective problem into a series of single-objective problems. Especially, an improved Benders decomposition approach is developed to solve each single-objective problem efficiently. Numerical experiments comprising an illustrative case and 200 random instances are conducted to evaluate the performance of proposed methods. Moreover, some managerial insights are drawn.

Model and solution approach to coordinate production-inventory strategies considering nonlinear price-sensitive demand: application to Canadian pulp and paper industry

This study addresses a practical problem within a multi-level supply chain where a wide range of customers can be served through different strategies such as make-to-stock, make-to-order, or vendor-managed inventory. The customer demand is stochastic, and sensitive to pricing associated with different production-inventory strategies. We propose a two-stage stochastic mixed-integer non-linear programming model. In the first stage, decisions are made regarding the selection of production-inventory strategies and pricing to maximise the expected profit. The second stage involves decisions related to production, inventory, and distribution, which are used to evaluate the first-stage decisions under various scenarios with different levels of accuracy. To solve the model, a metaheuristic approach based on the Simulated Annealing algorithm is developed. To showcase the practical applicability of our model and solution approach, we use a real case study in a Canadian pulp and paper supply chain. The results revealed that both the production-inventory strategy assigned to customers and the sales price underwent changes across scenarios. Furthermore, we demonstrated that by implementing the SA algorithm, we could improve the initial profit by up to 1.43% through slight adjustments in the sales price and assigned strategies for customers.

A multi-objective two-stage stochastic programming approach for wind farms and demand side provider decisions using the weighted metric method

The rapid expansion of wind-based renewable energy sources highlights the critical need to enhance reserve capacity within power systems. This research aims to develop an effective model for harnessing the combined potential of flexible demand-side resources, acting as demand-side providers (DSPs), to address the inherent unpredictability of wind power. The DSP framework comprises three demand response aggregators: the capacity market program (CAP), energy storage systems (ESSs), and plug-in electric vehicle parking lots (PEVs PLs). To achieve this, a multi-objective two-stage stochastic programming model is introduced. It not only minimizes operational costs for system operators but also maximizes returns for demand-side resource owners supervised by DSPs in the energy and reserve market environment. The problem is tackled using a mixed-integer linear programming (MILP) approach employing a weighted metric methodology. Within this framework, the Weibull probability distribution function (WPDF) is utilized to model wind power uncertainty, while the truncated Gaussian distribution function (GDF) encapsulates the uncertainty regarding electric vehicle owner behavior. The results demonstrate that employing this method for simultaneous programming of energy and reserve not only reduces system operator costs but also enhances profits for resource owners.

Optimization on Photovoltaics and Energy Storage Integrated Flexible Direct Current Distribution Systems of Buildings Considering Load Uncertainty Using Scenario Generation Method

This study explores the intricate challenge of energy demand uncertainty in the design of Photovoltaics and Energy Storage integrated Flexible Direct Current Distribution (PEDF) systems. Our objective is to examine the impact of different scenario generation methods on PEDF system optimization. We compare four approaches, including probabilistic techniques based on Monte Carlo simulation, Latin Hypercube Sampling for base scenario sampling, and a simulation-based method using building performance modeling. We evaluate these approaches using the Independent Scenario Optimization (ISO) and Two-Stage Stochastic Programming (TSSP) models, aiming to minimize the annual total cost within PEDF systems while addressing uncertainties. Our findings shed light on the optimal PEDF design under uncertainty, offering valuable insights for future decision-making in dynamic energy systems.

A Two-Stage Stochastic Programming Approach for the Design of Renewable Ammonia Supply Chain Networks

This work considers the incorporation of renewable ammonia manufacturing sites into existing ammonia supply chain networks while accounting for ammonia price uncertainty from existing producers. We propose a two-stage stochastic programming approach to determine the optimal investment decisions such that the ammonia demand is satisfied and the net present cost is minimized. We apply the proposed approach to a case study considering deploying in-state renewable ammonia manufacturing in Minnesota’s supply chain network. We find that accounting for price uncertainty leads to supply chains with more ammonia demand met via renewable production and thus lower costs from importing ammonia from existing producers. These results show that the in-state renewable production of ammonia can act as a hedge against the volatility of the conventional ammonia market.

Flexible resource optimal allocation strategy for distribution system resilience improvement under failure uncertainty

Extreme disasters have led to huge economical losses to the power system and posed a huge threat to the stable supply of electric energy. Especially for traditional passive distribution network systems, failures of transmission lines, towers, and other equipment will cause large-scale power outages and huge economic losses. In order to improve the ability of the power system to deal with extreme weather disasters, this paper proposes an optimal configuration strategy for the distribution network backup power resource oriented to resilience improvement, considering the uncertainty of equipment failure caused by disasters in the case of a known disaster range. The location and capacity of backup resources can be properly configured to reduce the load shedding of the power distribution network during the disaster process, thereby improving the resilience of the distribution network. The method proposed in this paper is modeled as a two-stage stochastic programming, which includes mix-integer constraints. In the first stage, the optimization problem confirms the capacity and location of the flexible resources. In the second stage, the operational constraints and costs of load shedding are considered. The resilience enhancement method is verified on the improved IEEE-33 node system, and the result shows that the method proposed in this paper can effectively enhance the resilience of the urban power network.

AI-enhanced robust method for integrated healthcare resource pre-positioning and patient scheduling

This paper aims to optimise healthcare resource pre-positioning, patient scheduling, and patient transferring under uncertain demands and stochastic resource consumption. We propose a two-stage stochastic programming model that formulates the patient scheduling problem as a Markov decision process. To address complexities and uncertainties, we use Artificial Intelligence (AI) techniques to improve both model formulation and algorithmic performance. To tackle the limited data challenge, we introduce a Wasserstein distance-based ambiguity set and propose a two-stage distributionally robust optimisation (DRO) model, which derives a deterministic equivalent using the Lagrangian dual of non-anticipativity constraints. The solution process is accelerated with a scenario decomposition approach and the K-means clustering method. Both theoretical and numerical results demonstrate the consistency of the two-stage DRO model with the sample average approximation (SAA) method. The potential of AI to improve the model's performance is evident through the significant reduction in computation time achieved with the K-means clustering approach without compromising solution quality. Compared to the SAA method, the DRO model exhibits a considerable reduction in the waiting penalty cost for out-of-sample cases, ranging from 43.06 % to 81.23 % . Numerical results show that the proposed algorithm outperforms SAA methods and several benchmark policies in terms of computational efficiency and solutions.

Attack graph-based stochastic modeling approach for enabling cybersecure semiconductor wafer fabrication

The increasing interconnectivity between information technology (IT) and operational technology (OT) has brought numerous benefits to advanced manufacturing, including the semiconductor industry, by enhancing the quality and by reducing the costs of wafer and integrated circuits (IC) fabrication. However, the transition of the semiconductor industry to secure digital manufacturing has also increased the importance of safeguarding networked assets from cybercriminals. To this end, a flexible scheme is developed in this paper to model wafer fabrication as a directed attack graph. This graph representation is then used as the basis of a two-stage stochastic programming model that considers uncertainty in attack probabilities while minimizing expected cyber risk by employing optimal, budget-constrained countermeasures (i.e., arc interdiction strategies). A sample average approximation coupled with a progressive hedging algorithm or Bender’s decomposition are proposed. Numerical experiments are performed to assess the viability of the proposed solution algorithms, the optimal arc interdiction strategies under various budget levels, and the effect of attack probability mappings. The results indicate that computing terminals nearest the manufacturing process have the highest defense priority and that the sample average approximation coupled with a progressive hedging algorithm is a suitable hybrid solution procedure for large instances when compared with sample average approximation coupled with Benders.

Demand Steering in a Last-Mile Delivery Problem with Home and Pickup Point Delivery Options

To increase the efficiency of last-mile delivery, online retailers can adopt pickup points in their operations. The retailer may then incentivize customers to steer them from home to pickup point delivery to reduce costs. However, it is usually uncertain whether the customer accepts this incentive to switch to pickup delivery. This setup gives rise to a new last-mile delivery problem with integrated incentive and routing decisions under uncertainty. We model this problem as a two-stage stochastic program with decision-dependent uncertainty. In the first stage, a retailer decides which customers to incentivize. However, customers’ reaction to the incentive is stochastic: they may accept the offer and switch to pickup point delivery, or they may decline the offer and stick with home delivery. In the second stage, after customers’ final delivery choices are revealed, a vehicle route is planned to serve customers via the delivery option of their choice. We develop an exact branch-and-bound algorithm and propose several heuristics to improve the algorithm’s scalability. Our algorithm solves instances with up to 50 customers, realizing on average 4%–8% lower last-mile delivery costs compared with the commonly applied approaches in the industry that do not use incentives or offer incentives to all customers. We also develop a benchmark policy that gives very fast solutions with a 2% average optimality gap for small instances and up to 2% average cost increase compared with the heuristic solutions. Funding: This project received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement [Grant 765395]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2023.0287 .

A Decomposition Algorithm for Two-Stage Stochastic Programs with Nonconvex Recourse Functions

A Decomposition Algorithm for Two-Stage Stochastic Programs with Nonconvex Recourse Functions

Multi-objective station-network synergy planning for regional integrated energy system considering energy cascade utilization and uncertainty

Regional integrated energy system (RIES) is conductive to integrating distributed renewable energy and represent a crucial form for constructing low-carbon energy systems in future cities. Addressing key issues in RIES planning such as station-network synergy optimization, uncertainty optimization, and multi-objective optimization, this paper proposes a synergy planning framework for RIES considering energy cascading utilization and uncertainty. Firstly, a dual-layer synergy planning model is established. In the upper layer, it resolves the station-network layout optimization problem considering user distribution and load complementarity. An improved P-median model is introduced to optimize the site selection and network layout of energy stations. In the lower layer, a two-stage stochastic programming model is proposed to address the configuration optimization problem for energy stations considering uncertainty. To enhance the energy utilization efficiency of RIES, exergy efficiency is incorporated into the multi-objective optimization model. A combination of ε-constrant methods and the TOPSIS method is employed for multi-objective solving and decision-making. The proposed models and methods are validated through a case study. The analysis of seven comparative experiments indicates that there is an optimal number of energy stations for the overall economic efficiency in RIES station-network layout optimization. Compared to single-objective optimization, multi-objective optimization considering both economic and exergy efficiency has advantages in improving energy utilization efficiency and reducing carbon emissions. When the load complementarity between multiple regions is higher, the interconnection gains between energy stations are greater. The research framework proposed in this study can provide essential references for designers of urban RIES.

Unified Branch-and-Benders-Cut for two-stage stochastic mixed-integer programs

Two-stage stochastic programs are a class of stochastic problems where uncertainty is discretized into scenarios, making them amenable to solution approaches such as Benders decomposition. However, classic Benders decomposition is not applicable to general two-stage stochastic mixed-integer programs due to the restriction that the second-stage variables should be continuous. We propose a novel Benders decomposition-based framework that accommodates mixed-integer variables in both stages as well as uncertainty in all of the recourse parameters. The proposed approach is a unified branch-and-Benders algorithm, where we use a heuristic to maintain a global upper bound and a post-processing phase to determine an optimal solution. This new approach is flexible, allowing practitioners to integrate acceleration techniques such as partial decomposition or convexification schemes. We demonstrate the efficiency of our approach versus classic ones on the stochastic server location problem, and its generality on a new, complex stochastic problem where the second stage is a traveling salesman problem.

Two-stage stochastic programming approach for fleet sizing and allocating rail wagon under uncertain demand

In this paper, a scenario-based two-stage stochastic programming model has been developed to determine the optimal number of wagons and allocation of full and empty wagons during different time periods to optimize profit. In allocating wagons, when the wagon arrives at a station, it is not possible to reallocate the wagon at the moment of arrival, this issue has not been addressed before. In addition, the possibility of renting wagons to/from other companies is concerned in the railway planning. These two actions have not been addressed before. In this paper, firstly, the research problem is formulated in the form of an integer programming model with deterministic data. Due to the inherent uncertainty in the transportation demands of the real case study, the deterministic model has been transformed into a scenario-based two-stage stochastic programming model with recourse. Then the L-shaped method was proposed to solve the stochastic model and by using it for 34 instances, its better efficiency than CPLEX software was shown. The stochastic model was solved using the L-shaped method and the results show that the studied company should rent 749 wagons to meet the demands, and on average, the stochastic model is expected to have about 1.1% added value in the company's profit compared to the deterministic model.

A two-stage stochastic programming model for bike-sharing systems with rebalancing

We study the problem of determining the target inventory level of stations in a bike-sharing system, when bikes can be rebalanced later during the day. We propose a two-stage stochastic programming formulation, where the target inventory decisions are made at the first stage, while the recourse decisions, related to rebalancing, are made at the second stage. In the literature, the problem of determining the target inventory levels is solved without taking into account the rebalancing problem, or these two problems are solved sequentially. We prove that more efficient bike-sharing systems can be obtained by integrating these two problems. Moreover, we show that our methodology provides better results than the deterministic formulation, and consider an effective matheuristic, based on the solution of the deterministic problem, to solve the stochastic program. Finally, we compare the solutions obtained by our approach with the actual allocation of bikes in the real bike-sharing system of the city of San Francisco. The results show the effectiveness of our approach also in a realistic setting.

Design of experiments for the stochastic unit commitment with economic dispatch models

We develop a Design and Analysis of the Computer Experiments (DACE) approach to the stochastic unit commitment problem for power systems with significant renewable integration. For this purpose, we use a two-stage stochastic programming formulation of the stochastic unit commitment-economic dispatch problem. Typically, a sample average approximation of the true problem is solved using a cutting plane method (such as the L-shaped method) or scenario decomposition (such as Progressive Hedging) algorithms. However, when the number of scenarios increases, these solution methods become computationally prohibitive. To address this challenge, we develop a novel DACE approach that exploits the structure of the first-stage unit commitment decision space in a design of experiments, uses features based upon solar generation, and trains a multivariate adaptive regression splines model to approximate the second stage of the stochastic unit commitment-economic dispatch problem. We conduct experiments on two modified IEEE-57 and IEEE-118 test systems and assess the quality of the solutions obtained from both the DACE and the L-shaped methods in a replicated procedure. The results obtained from this approach attest to the significant improvement in the computational performance of the DACE approach over the traditional L-shaped method.

“货到人”拣选系统的两阶段随机规划模型

“货到人”拣选系统的两阶段随机规划模型