Second-stage Decisions Research Articles

Ranking and selection with two-stage decision

Ranking & selection (R&S) is concerned with the selection of the best decision from a finite set of alternative decisions when the outcome of the decision has to be estimated using stochastic simulation. In this paper, we extend the R&S problem to a two-stage setting where after a first-stage decision has been made, some information may be observed and a second-stage decision then needs to be made based on the observed information to achieve the best outcome. We then extend two popular single-stage R&S algorithms, expected value of information (EVI) and optimal computing budget allocation (OCBA), to efficiently solve the new two-stage R&S problem. We prove the consistency of the new two-stage EVI (2S-EVI) and OCBA (2S-OCBA) algorithms. Experiment results on benchmark test problems and a two-stage multi-product assortment problem show that both algorithms outperform applying single-stage EVI and OCBA in the two-stage setting. Between 2S-EVI and 2S-OCBA, numerical results suggest that 2S-EVI tends to perform better with smaller number of decisions at first and second stage while 2S-OCBA has better performance for larger problems.

Gate appointment design in a container terminal: A robust optimization approach

Gate congestion is a main challenge faced by container terminals worldwide. In the scenario of gate congestion, the long waiting line of container trucks can block the traffic into the terminal, leading to disrupted terminal operations and unsatisfactory operating efficiency. In this paper, we study a gate appointment design problem that aims to manage the number of trucks allowed to a container terminal over time, so as to minimize the length of the truck waiting line and alleviate gate congestion. To optimize the gate appointment strategy, the truck service plans need also be devised, which involve the decisions for assigning trucks to yard blocks for container transhipment. We develop a two-stage robust optimization model for the problem by taking into account the uncertain service capacities of the yard blocks. Our model considers two decision stages, where the first-stage decision allocates slot capacities for the appointment slots, whereas the second-stage decision designs truck service plans with observed yard service capacity. We solve the two-stage robust model using an adapted column-and-row generation algorithm. In each iteration of the adapted column-and-row generation algorithm, the second-stage decision is generated by means of a pseudo-polynomial-time dynamic programming algorithm, while the first-stage decision is generated by a tailored scenario decomposition method. We test the computational performance of the proposed solution method on instances generated from real terminal operation data, and reveal managerial insights that would inspire terminal operators in the management of truck appointment for gate congestion mitigation.

Green retailer: A stochastic bi-level approach to support investment decisions in sustainable energy systems

This paper presents a bi-level approach to support retailers in making investment decisions in renewable-based systems to provide clean electricity. The proposed model captures the strategic nature of the problem and combines capacity sizing decisions for installed technologies with pricing decisions regarding the electricity tariffs to offer to a reference end-user, representative of a class of residential prosumers. The interaction between retailer and end-user is modeled using the Stackelberg game framework, with the former acting as a leader and the latter as follower. The reaction of the follower to the electricity tariff affects the retailer’s profit, which is calculated as the difference between the revenue generated from selling electricity and the total investment, operation and management costs. To account for uncertainty in wholesale electricity prices, renewable resource availability and electricity request, the upper-level problem is formulated as a two-stage stochastic programming model. First-stage decisions refer to the sizing of installed technologies and electricity tariffs, whereas second-stage decisions refer to the operation and management of the designed system. The model also incorporates a safety measure to control the average profit that can be achieved in a given percentage of worst-case situations, thus providing a contingency against unforeseen changes. At the lower level, the follower reacts to the offered tariffs by defining the procurement plan in terms of energy to purchase from the retailer or potential competitors, with the final aim of minimizing the expected value of the electricity bill. A tailored approach that exploits the specific problem structure is designed to solve the proposed formulation and extensively tested on a realistic case study. The numerical results demonstrate the efficiency of the proposed approach and validate the significance of explicitly dealing with the uncertainty and the importance of incorporating a safety measure.

Convex approximations of two-stage risk-averse mixed-integer recourse models

AbstractWe consider two-stage risk-averse mixed-integer recourse models with law invariant coherent risk measures. As in the risk-neutral case, these models are generally non-convex as a result of the integer restrictions on the second-stage decision variables and hence, hard to solve. To overcome this issue, we propose a convex approximation approach. We derive a performance guarantee for this approximation in the form of an asymptotic error bound, which depends on the choice of risk measure. This error bound, which extends an existing error bound for the conditional value at risk, shows that our approximation method works particularly well if the distribution of the random parameters in the model is highly dispersed. For special cases we derive tighter, non-asymptotic error bounds. Whereas our error bounds are valid only for a continuously distributed second-stage right-hand side vector, practical optimization methods often require discrete distributions. In this context, we show that our error bounds provide statistical error bounds for the corresponding (discretized) sample average approximation (SAA) model. In addition, we construct a Benders’ decomposition algorithm that uses our convex approximations in an SAA-framework and we provide a performance guarantee for the resulting algorithm solution. Finally, we perform numerical experiments which show that for certain risk measures our approach works even better than our theoretical performance guarantees suggest.

Two-stage hybrid model for supplier selection and order allocation considering cyber risk

In the context of collaborative manufacturing, cyber risk caused by cyber attacks may lead to severe supply chain disruption. Currently, supplier selection and order allocation is regarded as effective means to mitigate the risks that might cause disruption. Thus, we propose a two-stage hybrid model for supplier selection and order allocation under cyber risk. The hybrid model consists of fuzzy analytical hierarchy process (Fuzzy AHP), Technique for Order Preference by Similarity to Ideal Solution (TOPSIS), and two-stage mixed integer linear programming (MIP). Based on the extracted cyber risk indicators, a Fuzzy AHP is used to calculate the level of cyber risk of suppliers. TOPSIS is utilized to quantitatively evaluate the cyber risk of suppliers and determine the ranking of suppliers. Then, a two-stage MIP model is developed to support decision-making on order allocation. The first-stage decisions are determined without emergencies, and the second-stage decisions are determined under emergencies. The results reveal that application of the proposed two-stage hybrid model could mitigate the negative impacts of cyber risks. By providing a theoretical basis and quantitative method for cyber risk evaluation, this research is of theoretical and practical significance to the field of supply chain management.

Distributed surgical scheduling across collaborating hospitals considering stochastic duration and emergency demand

In this paper, we address the need for distributed surgical scheduling across multiple hospitals and multiple days. We develop an effective ex-ante distributed schedule to determine elective service operation and elective patient assignment plan in a hospital alliance with the goal of minimizing the total expected cost considering possible realizations of treatment durations and emergency demands. The problem is formulated into a two-stage stochastic integer programming where uncertainty is characterized by a scenario set. Cancellations of scheduled elective patients and inter-hospital transfers of materialized emergency patients are explicitly modeled as second-stage decisions. To solve the problem, we propose an integer decomposition algorithm that incorporates a novel logic-based optimality cut along with a validity proof. Additionally, we introduce several enhancements to improve the algorithm. Through numerical experiments, we demonstrate that our proposed algorithm outperforms both the commercial solver and the existing integer decomposition algorithm for the proposed problem. It provides effective solutions that closely approach the best upper bounds achieved by the solver in less time, or even surpass the solver’s best upper bounds. Furthermore, we provide operational insights into other commonly-used scheduling approaches through numerical comparisons, including the strengths and limitations of the most common way of scheduling elective patients and two strategies for handling emergencies within a hospital alliance.

Robust homecare service capacity planning

We study a home healthcare planning problem under the demand uncertainty, where the service type (authorization) and capacity are the first-stage decision and the homecare resource allocation is the second-stage decision which adapts to the demand realizations. We model the problem using an adaptive robust optimization technique where we construct a budget uncertainty set of demand using the well-known Mahalanobis Distance. We analyze the impact of the authorization, capacity decisions as well as the budget (robustness level) of the Mahalanobis uncertainty set onto the worst-case revenue. To solve the model, we develop a Benders decomposition algorithm that solves a pair of a mixed-integer second-order cone program (MISOCP) and a mixed integer linear program (MILP) in each iteration, both can be handled by off-the-shelf MIP solvers, with finite-step convergence. We also develop an affine approximation approach that directly solves one instance of MISOCP. Finally, sufficient numerical studies demonstrate the effectiveness of our model and the solution approaches.

A solar-powered bus charging infrastructure location problem under charging service degradation

Photovoltaic and energy storage system (PESS) offers a compelling pathway towards boosting green transportation due to its low carbon emissions. This study investigates a solar-powered bus charging infrastructure location problem by considering PESS. A two-stage robust optimization model is formulated to handle the uncertainty of charging service degradation. The first-stage decision is to determine which bus depots are to be upgraded with PESS. The second-stage decision is to conduct emergent bus and energy scheduling when the charging service degrades. Two objectives are optimized simultaneously. The first objective is to maximize the net benefits of PESS during day-to-day operations. The second objective is to minimize the unmet passenger demands during the charging service degradation. We implement a case study using a sub-network of Beijing public transport. The results present pieces of evidence that PESS can lower the daily bus charging costs and improve the service capacity of passengers when the charging service degrades.

A framework for optimal recruitment of temporary and permanent healthcare workers in highly uncertain environments

There has been a significant increase in the demand for temporary skilled workers in the health sector. They provide volume flexibility, but are generally more expensive than their permanent counterparts. In this paper, we propose a two-stage stochastic optimization framework to inform recruitment decision making for a period of highly uncertain demand in a setting where all patients must be served. The first stage identifies the number of permanent positions to advertise, and the second stage determines the number of temporary workers to recruit. Our framework accounts for the uncertainty in the permanent recruitment process, stochasticity of the service delivery, and asymmetry in demand information at the times of permanent and temporary recruitment. Under a general setting of the problem, we characterize the optimal first- and second-stage decisions analytically, propose fast numerical methods for finding their values, and prove some of their monotonicity properties. A case study based on data from a geriatric ward illustrates the application of our framework, and numerical experiments provide further managerial insights.

Designing a multi-period and multi-product resilient mixed supply chain network under chain-to-chain competition

PurposeThis study addresses resilient mixed supply chain network design (SCND) and aims to minimize the expected total cost of the supply chain (SC) considering disruptions. The capacity of facilities is considered uncertain. In order to get closer to real-world situations, competition between SCs is considered.Design/methodology/approachA two-stage stochastic programming model is developed for designing the SC network. The location of facilities and selection of suppliers are considered first-stage decisions, and the determination of materials and products flows are second-stage decisions. Some resilience strategies are applied to mitigate the negative impacts of disruptions.FindingsThe results indicate that considering resilience and applying the related strategies are vitally important, and resilience strategies can significantly improve the SC objective and maintain market share. Also, it is confirmed that unrealistic decisions will be made without considering the competition.Originality/valueThis study contributes to the literature by proposing a novel mathematical model for the resilient mixed SCND problem. The other contribution is considering the chain-to-chain competition in collecting returned products and selling recycled products to other SCs in a mixed SC under disruptions. Also, a novel hybrid metaheuristic is developed to cope with the complexity of the model.

Stochastic energy community trading model for day-ahead and intraday coordination when offering DER’s reactive power as ancillary services

A two-stage stochastic programming energy trading model is presented in this article to measure the distributed energy resources’ capability to provide reactive power as ancillary services from an energy community to the distribution system operator. The formulation proposed models the day-ahead and the intraday markets as first and second-stage decisions, respectively, using the second-order cone relaxation of the optimal power flow to represent the network limitations. In addition, the model considers that the energy community trades energy operating under a collaborative scheme and minimizing the global cost. Likewise, the formulation (i) includes community batteries to study the effect on the total cost, (ii) identifies the extra energy charged and discharged by the agents’ batteries to face the uncertainty related to the intraday market, as an agents’ flexibility service for the community, and (iii) prevents the simultaneous buying and selling of energy in the local energy market by an agent. The model has been programmed in Python-Pyomo and tested in three radial distribution systems under three different scenarios showing that the DERs can (i) self-satisfy the reactive power demand, (ii) provide reactive power to the DSO, and (iii) decrease the community cost significantly in an eventual ancillary services market.

Municipal solid waste recycling network with sustainability and supply uncertainty considerations

• A stochastic optimization model is developed by explicitly considering sustainability • The developed two-stage stochastic MILP model optimizes the MSW recycling network • The fuzzy best-worst method is used to extend the social life cycle assessment method • The environmental cost is controlled utilizing a well-known risk measure • MRFs, Incinerator, Composting, and LFGRS are sustainable facilities for Mazandaran The stochastic nature of some parameters in a municipal solid waste (MSW) management system justifies the use of stochastic programming models. In this research, a multi-period two-stage stochastic model is formulated as a mixed-integer linear programming model for optimal and sustainable use of MSW by explicitly considering the social, economic, and environmental dimensions of sustainability. The decision variables are categorized into two groups. The first-stage decision variables determine the location, capacity, and type of waste processing plants and facilities. The second-stage decision variables consider transportation, allocation, and distribution of wastes and products. The objective function of the proposed model minimizes various costs of the system. The environmental cost is controlled utilizing a well-known risk measure and to consider the social impacts, the Social Life Cycle Assessment (S-LCA) approach is extended using the fuzzy best-worst method. To handle the uncertainty in the amount of generated waste, a backward scenario reduction method is simulated. Mazandaran province in Iran is selected as a real-life case study to evaluate the performance of the applied method. The assessment of the robustness of the stochastic solution illustrates the significance of using the stochastic model instead of its deterministic one.

Integrated berth and yard space allocation under uncertainty

Improving container ports’ operational efficiency to reduce the delays at ports concerns global port operators significantly. Optimizing the tactical level allocation of quay side and yard side resources to vessel calls is a typical lever to mitigate the delays in operations. Another lever is by strategical level planning that re-adjust the preferred visiting time windows of vessel calls. It aims to absorb the delays at the scheduling stage rather than in the operations stage. This paper integrates the planning and operations at container ports to jointly optimize strategical level planning and tactical level berth and yard space allocation under uncertain vessel arrival times and uncertain numbers of loading/unloading containers. The problem is formulated as a two-stage stochastic integer programming model. To solve it, we develop an original decomposition algorithm that passes columns of second-stage problems to the first-stage problem to approximate the second-stage decision making. Numerical experiments are conducted to validate the effectiveness and efficiency of our proposed algorithm. Some managerial implications for port operators are also obtained.

Modeling the risk-based decisions of the microgrid in day-ahead energy and reserve markets considering stochastic dispatching of electrical and thermal energy storages

• The MGO’s bids in the markets are optimized with dispatching of local resources. • Uncertainties of electrical demand and power generation of PV are modeled. • A robust two-stage stochastic optimization model is developed. • The most capacity of electrical energy storage is used to provide reserve capacity. • With increasing the robust parameter, the MGO’s cost increases. In this paper, the electrical and thermal energy management problem of a micro-grid operator (MGO) is addressed under uncertainties aiming at participating in the day-ahead energy and reserve markets. For this purpose, a robust two-stage stochastic model is developed to protect the first stage MGO’s decisions, i.e., its bids in the energy and reserve markets, against the uncertainty of the real-time energy market price. This is done through stochastic dispatching of the MG resources which includes the electrical and thermal energy storages and the combined heat and power unit as the second-stage decisions. The results showed that the MGO’s expected total cost decreases when it participates in both the energy market and the reserve market in comparison with the case it only participates in the energy market. Also, the risk-based behavior of the MGO showed that increasing the robust parameter decreases the reserve provided for the market and the net power trading with the market. However, the proposed robust two-stage stochastic model leads to a smaller reduction of the MGO’s first-stage decisions in the worst case in comparison with the conventional methods, i.e. deterministic and probabilistic ones. This issue proves the effectiveness of the proposed approach to protect the MGO’s decisions against the uncertainties.

Robustly Coordinated Generation Dispatch and Load Shedding for Power Systems Against Transient Instability Under Uncertain Wind Power

Transient stability of a power system can be significantly affected by wind power generators due to their stochastic power output and complex dynamic characteristics. This paper proposes a robust optimization approach for coordinating generation dispatch and emergency load shedding against transient instability under uncertain wind power output. The problem is modelled as a two-stage robust optimization (TSRO) model considering transient stability constraints, where the first-stage is to optimize the generation dispatch (preventive control) before a contingency and the second-stage decision is the emergency load shedding (emergency control) after the contingency occurrence under the worst case of wind power variation. To solve this TSRO problem, this paper also proposes a solution algorithm which integrates transient stability assessment and transient stability constraint construction in a column and constraint generation framework. The proposed method is validated on the New-England 39-bus system and the Nordic32 system, which shows high computational efficiency and stability robustness against uncertain wind power.

Short-Term nurse schedule adjustments under dynamic patient demand

We study two-stage short-term staffing adjustments for the upcoming nursing shift. Our proposed adjustments are first used at the beginning of each 4-hour nursing shift, shift t, for the upcoming shift, shift t + 1. Then, after observing actual patient demand for nursing at the start of shift t + 1, we make our final staffing adjustments to meet the patient demand. We model six different adjustment options for the two-stage stochastic programming model, five options available as first-stage decisions and one option available as the second-stage decision. We develop a two-stage stochastic integer programming model, which minimizes total nurse staffing costs and the cost of adjustments to the original schedules, while ensuring the coverage of nursing demand. Our experimental results, using the data from an urban Children’s Hospital, indicate that the developed stochastic nurse schedule adjustment model can deliver cost savings up to 18% for the medical units, compared to alternative no short-term adjustment scheduling models. The proposed stochastic adjustments model successfully keeps average understaffing percentages under 2% throughout the staffing horizon.

A STOCHASTIC OPTIMIZATION MODEL FOR THE IRREGULAR KNAPSACK PROBLEM WITH UNCERTAINTY IN THE PLATE DEFECTS

The present research deals with the two-dimensional knapsack problem by considering the cutting of irregular items from a rectangular plate with defects. While the defects are only known at the time of cutting (in the future), we need first to select which items to produce from cutting the plate. The final items cannot have any defects and the goal is to maximize the profit from cutting the plate and producing the items. We propose a two-stage stochastic optimization model that makes use of a discrete set of scenarios with the realization of the plate defects. The first-stage decisions involve selecting items for cutting and possible production. The second-stage decisions consider the positioning of items in the plate given the scenarios with defects, and then the cancellation and non-production of some selected items, if any. We also extend this model to include a measure of risk, aiming at robust solutions. We perform computational tests on instances adapted from the literature that consider three types of defects, eight scenarios, and four cases for determining each scenario’s probability. The tests evaluate the impact of uncertainties on the problem by calculating the expected value of perfect information and the value of the stochastic solution. The results indicate a percentage reduction in the profit of up to 27.7%, on average, when considering a fully risk-averse decision-maker.

Lot-Size Models with Uncertain Demand Considering Its Skewness/Kurtosis and Stochastic Programming Applied to Hospital Pharmacy with Sensor-Related COVID-19 Data.

Governments have been challenged to provide timely medical care to face the COVID-19 pandemic. Under this pandemic, the demand for pharmaceutical products has changed significantly. Some of these products are in high demand, while, for others, their demand falls sharply. These changes in the random demand patterns are connected with changes in the skewness (asymmetry) and kurtosis of their data distribution. Such changes are critical to determining optimal lots and inventory costs. The lot-size model helps to make decisions based on probabilistic demand when calculating the optimal costs of supply using two-stage stochastic programming. The objective of this study is to evaluate how the skewness and kurtosis of the distribution of demand data, collected through sensors, affect the modeling of inventories of hospital pharmacy products helpful to treat COVID-19. The use of stochastic programming allows us to obtain results under demand uncertainty that are closer to reality. We carry out a simulation study to evaluate the performance of our methodology under different demand scenarios with diverse degrees of skewness and kurtosis. A case study in the field of hospital pharmacy with sensor-related COVID-19 data is also provided. An algorithm that permits us to use sensors when submitting requests for supplying pharmaceutical products in the hospital treatment of COVID-19 is designed. We show that the coefficients of skewness and kurtosis impact the total costs of inventory that involve order, purchase, holding, and shortage. We conclude that the asymmetry and kurtosis of the demand statistical distribution do not seem to affect the first-stage lot-size decisions. However, demand patterns with high positive skewness are related to significant increases in expected inventories on hand and shortage, increasing the costs of second-stage decisions. Thus, demand distributions that are highly asymmetrical to the right and leptokurtic favor high total costs in probabilistic lot-size systems.

Adaptive robust vulnerability analysis of power systems under uncertainty: A multilevel OPF-based optimization approach

With the growing level of uncertainties in today’s power systems, the vulnerability analysis of a power system with uncertain parameters becomes a must. This paper proposes a two-stage adaptive robust optimization (ARO) model for the vulnerability analysis of power systems. The main goal is to immunize the solutions against all possible realizations of the modeled uncertainty. In doing so, the uncertainties are defined by some pre-determined intervals defined around the expected values of uncertain parameters. In our model, there are a set of first-stage decisions made before the uncertainty is revealed (attacker decision) and a set of second-stage decisions made after the realization of uncertainties (defender decision). This setup is formulated as a mixed-integer trilevel nonlinear program (MITNLP). Then, we recast the proposed trilevel program to a single-level mixed-integer linear program (MILP), applying the strong duality theorem (SDT) and appropriate linearization approaches. The efficient off-the-shelf solvers can guarantee the global optimum of our final MILP model. We also prove a lemma which makes our model much easier to solve. The results carried out on the IEEE RTS and modified Iran’s power system show the performance of our model to assess the power system vulnerability under uncertainty.

The effect of expert recommendations on intergovernmental decision-making: North Korea, Iran, and non-proliferation sanctions in the Security Council

The article explores whether and to what extent expert recommendations affect decision-making within the Security Council and its North Korea and Iran sanctions regimes. The article first develops a rationalist theoretical argument to show why making many second-stage decisions, such as determining lists of items under export restrictions, subjects Security Council members to repeating coordination situations. Expert recommendations may provide focal point solutions to coordination problems, even when interests diverge and preferences remain stable. Empirically, the article first explores whether expert recommendations affected decision-making on commodity sanctions imposed on North Korea. Council members heavily relied on recommended export trigger lists as focal points, solving a divisive conflict among great powers. Second, the article explores whether expert recommendations affected the designation of sanctions violators in the Iran sanctions regime. Council members designated individuals and entities following expert recommendations as focal points, despite conflicting interests among great powers. The article concludes that expert recommendations are an additional means of influence in Security Council decision-making and seem relevant for second-stage decision-making among great powers in other international organisations.