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In this paper, we consider a project-based organization that deals with an assignment problem in which a set of projects must be assigned to a group of project managers. This assignment is done based on the relative contributions of projects to the organizational mission and a matching score between each pair of a project and a project manager. We assume that some projects are deterministic, and the organization has signed their corresponding contracts while others are stochastic, i.e. the organization has submitted bids for these projects and may or may not win them in the future. Furthermore, we consider a finite planning horizon and presume a predetermined start time for each deterministic and stochastic project. We develop two models including a multi-stage stochastic integer programming model and a stochastic dynamic programming model to solve the problem. The latter shows better performance for small-size and less complex instances whereas the former gives better performance for more complex instances. We also developed a heuristic algorithm to solve large-size and more complex instances. Computational results indicate that the developed heuristic algorithm can reach near-optimal solutions in reasonable CPU run times and dominates the two other solution approaches particularly for large-size instances. [ABSTRACT FROM AUTHOR]

Aimed at sufficiently utilizing available data and prior distribution information, we introduce a data-driven Bayesian-type approach to solve multi-stage convex stochastic optimization, which can easily cope with the uncertainty about data process's distributions and their inter-stage dependence. To unravel the properties of the proposed multi-stage Bayesian expectation optimization (BEO) problem, we establish the consistency of optimal value functions and solutions. Two kinds of algorithms are designed for the numerical solution of single-stage and multi-stage BEO problems, respectively. A queuing system and a multi-stage inventory problem are adopted to numerically demonstrate the advantages and practicality of the new framework and corresponding solution methods, compared with the usual formulations and solution methods for stochastic optimization problems. [ABSTRACT FROM AUTHOR]

Abstract The recent pandemic caused by COVID-19 is considered an unparalleled disaster in history. Developing a vaccine distribution network can provide valuable support to supply chain managers. Prioritizing the assigned available vaccines is crucial due to the limited supply at the final stage of the vaccine supply chain. In addition, parameter uncertainty is a common occurrence in a real supply chain, and it is essential to address this uncertainty in planning models. On the other hand, blockchain technology, being at the forefront of technological advancements, has the potential to enhance transparency within supply chains. Hence, in this study, we develop a new mathematical model for designing a COVID-19 vaccine supply chain network. In this regard, a multi-channel network model is designed to minimize total cost and maximize transparency with blockchain technology consideration. This addresses the uncertainty in supply, and a scenario-based multi-stage stochastic programming method is presented to handle the inherent uncertainty in multi-period planning horizons. In addition, fuzzy programming is used to face the uncertain price and quality of vaccines. Vaccine assignment is based on two main policies including age and population-based priority. The proposed model and method are validated and tested using a real-world case study of Iran. The optimum design of the COVID-19 vaccine supply chain is determined, and some comprehensive sensitivity analyses are conducted on the proposed model. Generally, results demonstrate that the multi-stage stochastic programming model meaningfully reduces the objective function value compared to the competitor model. Also, the results show that one of the efficient factors in increasing satisfied demand and decreasing shortage is the price of each type of vaccine and its agreement.

Variations of physical and chemical characteristics of biomass reduce equipment utilization and increase operational costs of biomass processing. Biomass processing facilities use sensors to monitor the changes in biomass characteristics. Integrating sensory data into the operational decisions in biomass processing will increase its flexibility to the changing biomass conditions. In this paper, we propose a multi-stage stochastic programming model that minimizes the expected operational costs by identifying the inventory level and creating an operational decision policy for equipment speed settings. These policies take the sensory information data and the current biomass inventory level as inputs to dynamically adjust inventory levels and equipment settings according to the changes in the biomass' characteristics. We ensure that a prescribed target reactor utilization is consistently achieved by penalizing the violation of the target reactor feeding rate. A case study is developed using real-world data collected at Idaho National Laboratory's biomass processing facility. We show the value of multi-stage stochastic programming from an extensive computational experiment. Our sensitivity analysis indicates that updating the infeed rate of the system, the processing speed of equipment, and bale sequencing based on the moisture level of biomass improves the processing rate of the reactor and reduces operating costs. [ABSTRACT FROM AUTHOR]

Production planning in maintenance facilities that rely on the availability of certified operators is a challenging task. This is primarily attributed to the need for the integration of workforce training with the operations schedule, in addition to the uncertain repair time of some components. We confine our attention to the repair and overhaul of advanced technological devices that are composed of several components. In this context, this study proposes a novel approach to embedding uncertain repair times of faulty components and develops a multi-stage stochastic programming model for integrated production and workforce planning in such facilities under independent random repair time. Our numerical experiments conducted on a range of problem instances expose the significant value of incorporating repair time uncertainty in this problem. [ABSTRACT FROM AUTHOR]

Transforming Pulp and Paper (P&P) mills into Integrated Forest Biorefineries (IFBR) is a prominent solution to save Canadian P&P industry that has been facing decline of conventional paper demand. We propose a comprehensive decision model for the design of IFBR value chains by taking the uncertain demand of bioproducts into consideration. In particular, we propose a multi-stage stochastic programming model to obtain the optimal investment plan over a long-term planning horizon in the presence of various market trends. We also develop a Monte-Carlo simulation platform to validate the proposed model and to compare its performance with alternative decision models. The model is applied to a realistic case study inspired from P&P companies in Canada, where the value of incorporating the dynamic nature of uncertain demand has been estimated. Further, we elaborate on the value of considering flexibility in terms of adjusting the investment plan in response to changes in the demand. Our results indicate that the demand for bioproducts has a substantial impact on the profitability of the IFBR. We also demonstrated the significant value of explicitly incorporating the uncertainty in IFBR network design as well as adapting the investment plan to the changes in the demand. [ABSTRACT FROM AUTHOR]

Abstract The carbon capture and storage (CCS) technique gains much attention due to its role in reducing CO2 emissions. By introducing flexible CCS devices into conventional power plants, the low‐carbon economic operation of the power system can be achieved. However, the uncertainty of renewable energy makes it hard to obtain an optimal operation policy. First, the detailed model of CCS consisting of bypass venting stacks, solvent storage tanks, absorber, and stripper is described. Then, a low‐carbon economic dispatch model of the power system with CCS is proposed to minimize the total operation and renewable energy abandonment penalty costs. Next, the problem is reformulated as multi‐stage stochastic programming, and the stochastic dual dynamic programming (SDDP) method is applied to relieve the computation burden. To control the probability of stopping the algorithm prematurely, a new stopping criterion based on a two‐sided hypothesis test is proposed. Finally, the effectiveness of the proposed model and the new stopping criterion is demonstrated by case studies on a practical power system.

Supply chains are one of the key tools in optimizing production and distribution simultaneously. However, information uncertainty is always a challenge in production and distribution management. The main purpose of this paper is to design a two-echelon supply chain in a multi-cycle state and in conditions of demand uncertainty. The task includes determining the number and location of distribution centers, planning capacity for active distribution centers, and determining the amount of shipments between different levels so that the total costs of the chain are minimized. Uncertainty is applied through discrete scenarios in the model and the problem is formulated by multi-stage stochastic programming method in the form of a mixed integer linear model. The results acquired using two indicators called VMS and VSS demonstrated that modeling the supply chain design problem with the multi-stage stochastic approach can result in significant costs reduction. Plus, utilizing mathematical expectation can generate misleading results, therefore resulting in the development of supply chain designs incapable of satisfying demand due to its overlooked limitations.

The carbon capture and storage (CCS) technique gains much attention due to its role in reducing CO2 emissions. By introducing flexible CCS devices into conventional power plants, the low‐carbon economic operation of the power system can be achieved. However, the uncertainty of renewable energy makes it hard to obtain an optimal operation policy. First, the detailed model of CCS consisting of bypass venting stacks, solvent storage tanks, absorber, and stripper is described. Then, a low‐carbon economic dispatch model of the power system with CCS is proposed to minimize the total operation and renewable energy abandonment penalty costs. Next, the problem is reformulated as multi‐stage stochastic programming, and the stochastic dual dynamic programming (SDDP) method is applied to relieve the computation burden. To control the probability of stopping the algorithm prematurely, a new stopping criterion based on a two‐sided hypothesis test is proposed. Finally, the effectiveness of the proposed model and the new stopping criterion is demonstrated by case studies on a practical power system. [ABSTRACT FROM AUTHOR]

Given the increasing population growth and rapid global economic development, the conflict between reduced available water resources and increased water demand in many countries has become a challenging issue; confirming the necessity of allocating optimal water resources to equate water conservation and environmental sustainability. Uncertainties and climate change together deteriorate the complexity of optimal allocation related to water resources used for the agricultural sector. The present study sought to design an uncertainty-based random boundary interval multi-stage stochastic programming (URBIMSP) model by integrating the Kataoka's criterion method and chance-constrained programming within a general optimization framework to control the uncertain information which exists in both constraints and goal functions. The developed model was applied to a case study of water resources allocation in the Marand basin, Iran. Then, a conjunctive water allocation system was arranged to identify the introduced model in two planning periods which were 2023 to 2025 and 2026 to 2028. The results indicated that different water allocation schemes and water shortages with varying net system benefits would be achieved under different stream flow levels and different risk scenarios. The result was significant for carrying out a comprehensive analysis of the tradeoffs between total net benefits and related risks. Finally, the findings demonstrated that improving irrigation efficiency and upgrading irrigation systems are the key strategies for farmers to achieve economic benefits while also help to mitigate environmental impacts. [ABSTRACT FROM AUTHOR]

The recent COVID-19 pandemic revealed that healthcare networks must have a flexible and effective structure. In this study, we develop a viable healthcare network design for a pandemic using a multi-stage stochastic approach. We propose a multi-level network that includes health centers, computed tomography scan centers, hospitals, and clinics. Patients have conditions to returning to normal life or quarantining at home. Three objectives are defined: maximizing the probability of patient recovery, minimizing the costs of all centers in the network, and minimizing the Coronavirus death rate. We investigate a real case study in Iran to demonstrate the model's applicability. Finally, we compare the healthcare supply chain network design in a pandemic with a normal situation to advise how the network can continue to remain viable. [ABSTRACT FROM AUTHOR]

The multi-stage stochastic programming (MSP) approach is widely used to solve financial planning problems owing to its flexibility. However, the size of an MSP problem grows exponentially with the number of stages, and such problem can easily become computationally intractable. Financial planning problems often consider planning horizons of several decades, and thus, the curse of dimensionality can become a critical issue. Stochastic dual dynamic programming (SDDP), a sampling-based decomposition algorithm, has emerged to resolve this issue. While SDDP has been successfully implemented in the energy domain, few applications of SDDP are found in the finance domain. In this study, we identify the major obstacle in using SDDP to solve financial planning problems to be the stagewise independence assumption and propose a partially observable SDDP (PO-SDDP) framework to overcome such limitations. We argue that the PO-SDDP framework, which models uncertainties using discrete-valued partially observable Markov states and introduces feasibility cuts, can properly address large-scale financial planning problems. [ABSTRACT FROM AUTHOR]

Food waste and proper methods to deal with it are one of the main challenges of supply chain network management. The majority of studies on how to use mathematical models in the supply chain have focused on goods that are at their peak of freshness as soon as they are produced and deteriorate over time. While some products experience an increase in value at the start of their life cycle, this value eventually reaches its maximum level, and after this point, these products experience a decline in value before being eliminated from the consumption cycle. The objective of this study is to develop a comprehensive inventory–routing model suitable for supply chain networks where products exhibit an increase and decrease in value over time. By considering the randomness and dynamic uncertainty of market demands and the fact that each period has effects on the next period, The proposed model employs a multi-stage stochastic programming (MSSP) approach. By doing so, the model ensures a balanced flow between different components of the network while considering nondeterministic demand under various scenarios that are shown in a tree of scenarios. The utilization of MSSP leads to more reliable solutions compared to deterministic models, making it possible for chain stores to make well-informed decisions in their inventory management and distribution strategies. Ultimately, this approach results in cost savings for chain stores handling such products. This research makes a significant contribution to the existing literature by demonstrating the effectiveness of the proposed model on actual data and highlighting the benefits of using stochastic programming in supply chain optimization. [ABSTRACT FROM AUTHOR]

The increased deployment of electricity-based hydrogen production strengthens the coupling of power distribution system (PDS) and hydrogen energy system (HES). Considering that power to hydrogen (PtH) has great potential to facilitate the usage of renewable energy sources (RESs), the coordination of PDS and HES is important for planning purposes under high RES penetration. To this end, this paper proposes a multi-stage co-planning model for the PDS and HES. For the PDS, investment decisions on network assets and RES are optimized to supply the growing electric load and PtHs. For the HES, capacities of PtHs and hydrogen storages (HSs) are optimally determined to satisfy hydrogen load considering the hydrogen production, tube trailer transportation, and storage constraints. The overall planning problem is formulated as a multistage stochastic optimization model, in which the investment decisions are sequentially made as the uncertainties of electric and hydrogen load growth states are revealed gradually over periods. Case studies validate that the proposed co-planning model can reduce the total planning cost, promote RES consumption, and obtain more flexible decisions under long-term load growth uncertainties.

Considering the importance of selecting suppliers based on the dimensions of sustainability in the supply chain, after identifying and selecting sustainability and risk criteria in accordance with Jey Oil Refining Company, by developing a multi-stage stochastic program and creating a risk constraint by the CVaR risk value criterion for quantitative criteria. Also, in terms of points calculated by FTOPSIS and FMEA methods for quality criteria, the optimal selection of suppliers, sourcing strategy and order allocation in a multi-period supply chain planning under operational risk and disruption were discussed. In order to reduce supply risk and achieve a flexible planning as a mitigation strategy, the option contract and the trading market were considered as two options to supply raw materials. The product demand, the market price of the materials, the purchase price and the apply price of the option contract, the supply quantity and the supply quantity of the option contract are random. To model uncertainty, discrete scenarios are generated through a simulation approach, and then, a scenario reduction method is used to construct a scenario tree. The application of the stochastic model, the performance of risk measurement policies, and the importance of mitigation strategies to provide some managerial insights have been investigated.

This article studies a model for risk aversion when designing a flexible capacity expansion plan for a multi-facility system. In this setting, the decision maker can dynamically expand the capacity of each facility given observations of uncertain demand. We model this situation as a multi-stage stochastic programming problem, and we express risk aversion through the Conditional Value-at-Risk (CVaR) and a mean-CVaR objective. We optimize the multi-stage problem over a tractable family of if–then decision rules using a decomposition algorithm. This algorithm decomposes the stochastic program over scenarios and updates the solutions via the subgradients of the function of cumulative future costs. To illustrate the practical effectiveness of this method, we present a numerical study of a decentralized waste-to-energy system in Singapore. The simulation results show that the risk-averse model can improve the tail risk of investment losses by adjusting the weight factors of the mean-CVaR objective. The simulations also demonstrate that the proposed algorithm can converge to high-performance policies within a reasonable time, and that it is also more scalable than existing flexible design approaches. [ABSTRACT FROM AUTHOR]

Given the vast, continuous and increasing changes in global markets and customer demand, the use of powerful and reliable tools to support decision makers is inevitable. For this purpose, in this research, we have tried to address the issue of production planning by considering the system dynamics and uncertainty in customer demand. The production system assumed in this model is a hybrid push-pull production system. By applying this approach, the proposed model is also adapted for a fully push or pull production system. Production planning is usually done in the medium term, and the decisions made at each stage of time will also affect future plans. Therefore, in the face of potential customer demand, the multi-stage stochastic programming method has been used in order to make decisions with a view to the entire time horizon. The purpose of the proposed model is to maximize profits through the optimal use of production capacity, appropriate pricing policies and material resource planning. Finally, the proposed model is examined through conventional numerical analyzes in the stochastic programming method. [ABSTRACT FROM AUTHOR]

Supporting wine production operations in an increasingly global market has grown ever more challenging. Export-focused wineries supply many foreign clients, often requiring different labels for the same kind of wine. Order forecasts tend to be highly inaccurate, and wineries must be able to quickly react to changes, making lot-sizing an important consideration. One tool to reduce product misallocation is postponing product differentiation, where the natural decoupling point for premium wine is the labelling process. However, the double handling involved incurs additional costs and time penalties. We analyse the performance impact of postponing the labelling of bottled wines by developing a multi-stage mixed-integer stochastic programming model with full recourse for demand scenarios. The underlying data and policies are based on an unnamed Chilean export-focused winery. The model supports lot-sizing under several winery production policies. We experiment with different levels of capacity tightness, demand variability and demand correlation between wines, optimising for reducing order backlogs, inventory levels and line set-ups. While we find that some amount of postponement will always be recommended, the exact mix and degree depend on these external factors. Postponement has the most benefits when production capacity is moderately tight, demand variability is high and wines have negatively correlated demands. [ABSTRACT FROM AUTHOR]

The problem of stock portfolio selection is one of the most critical problems in financial markets. The portfolio selection problem is to find an optimal solution to allocate a fixed amount of capital to a set of available stocks with the objective function of having the maximum expected rate of return and, at the same time, the least possible risk. Among the shortcomings that most stock portfolio models have is not paying attention to future changes and focusing too much on past information. This study aims to provide a framework that addresses some shortcomings and provides a practical tool. In this regard, this study suggests a multi-objective and multi-stage stochastic model for portfolio selection in the financial market. The stage refers to the periods in which the stock portfolio will be reviewed. By combining the scenario generation model with multi-stage stochastic programming, the investor is expected to achieve a suitable solution based on past information and various future scenarios. To solve the proposed model, a meta-heuristic algorithm whose main idea is derived from the harmony search algorithm is proposed. Finally, numerical instances were utilized by the use of real stock information from the Iranian stocks market. The algorithm proposed in this research was compared with a genetic algorithm in terms of the quality of the generated solutions and the runtime of the algorithms, and the superiority of the proposed algorithm has been proven. [ABSTRACT FROM AUTHOR]

We construct investment glide paths for a retirement plan using both traditional asset classes and deferred annuities (DAs). The glide paths are approximated by averaging the asset proportions of stochastic optimal investment solutions. The objective function consists of power utility in terms of secured retirement income from purchased DAs, as well as a bequest that can be withdrawn before retirement. Compared with conventional glide paths and investment strategies, our DA-enhanced glide paths provide the investor with higher welfare gains, more efficient investment portfolios and more responsive retirement income patterns and bequest levels to different fee structures and personal preferences. [ABSTRACT FROM AUTHOR]

As one of the essential commodities in the agricultural sector, rural farmers generally cultivate rice based on experience in agricultural fields. This method of cultivation has led to a waste of natural resources. In this study, the aim is to find suitable areas and determine the pattern of rice cultivation using a two-phase methodology. In the first phase, GIS integration and the best-worst method have been used to classify suitable areas for rice cultivation in Iran. The first phase's result is considered an input to the second phase, i.e., the optimization model to determine the pattern of rice cultivation. A multi-stage stochastic optimization approach has been used in the second phase to consider the weather uncertainty in all periods. Climatic conditions in each period are modeled in three scenarios. The application of the proposed model has been investigated in a case study in Iran. As a result, it has been observed that most of the suitable areas for rice cultivation are located in the north and western parts of Iran. Also, the suitable cultivation pattern for most rice farmers is the high-yield cultivation using the transplanting method.

Objective: Optimizing asset allocation at the asset class level and measuring the insolvency risk of the Social Security Organization (SSO)by considering the value at risk constraint. Methods: At first we hand-collect the book value of assets for the SSO using its financial statements from 2001 through 2015 and categorize assets into three asset classes: stocks, real estate and bonds. We then estimate the market value of assets using returns for the corresponding market during this period. Subsequently, we generate 300 forecasts for each market return over a 75-year horizon by the Monte Carlo simulation method. These efforts, Combined with the predictions of expenditures and contributions of the SSO from the International Labor Organization (ILO), for helping us have sufficient data to design and solve a multi-stage stochastic programming model. Also, we have adequate data to calculate the probability of insolvency in the SSO by counting the number of infeasible solutions through all forecasts in the model. Results: We obtain optimal allocation for all three asset classes and propose the types of reforms required for decreasing the insolvency risk in the SSO throughout the long term mending period in the way that imposes minimum pressure on pensioners and contributors simultaneously. Conclusion: Despite the constraints faced by the SSO in terms of asset allocation, our results indicate that certain allocations are capable of resolving the underfunding problem for the SSO over the studied horizon. This solution involves implementing a set of gradual parametric reforms and optimal asset allocation. In a way that could meet stakeholders' purpose. We consider the restriction that the state is facing in payback its debt to the SSO whereby securitize them. After all, legislation reforms in running an independent regulator organization are essential. These exercises make us impose an excellent government for pension funds and construct an investment policy for them to reduce harmful intervention

The planning process of wood remanufacturing operations encompasses challenging characteristic, including divergent co-production (one log tree may produce several different products), alternative processes (different receipts exist to produce the same products), short order cycle, dynamic market behaviour (with highly varying demand) and imperfect raw materials (due to its biological nature, the yield vary considerably). To deal with this complexity, in this paper random demand is modeled as scenario tree and three new predictive multi-stage stochastic programming models are developed with multiple objective functions. After implementing them employing datasets from a wood remanufacturing partner in Canada, the proposed models are compared to a reactive re-planning approach. The obtained results indicate that the new models exhibit higher quality solutions in comparison with their corresponding deterministic two-stage models. We also determine the number of stages for which the multi-stage programs provide better planning than the re-planning approach.

The large amount of carbon emitted due to industrial development is posing serious climate change. Under these circumstances, hydrogen energy is being used as a major energy source and is attracting attention as an alternative energy to fossil fuels, characterized by high carbon emissions. While the use of hydrogen has the obvious benefit of a low carbon footprint, the hydrogen production process possesses a high carbon footprint difference depending on the source of utilized electricity. This study considers several situations arising in an electrolysis-based hydrogen supply chain network (HSCN). A multi-stage stochastic programming (SP) is modeled to consider the uncertainty of demand and transportation capacity generated in HSCN. Moreover, a typical stochastic model has a chain rule between the decision variables resulting from nonlinearity and uncertainty, and these features make it difficult to derive an effective solution. Thus, in this study, a new methodology, Weighted Scenario Sample Average Approximation (WSSAA), is proposed to derive an effective solution for the multistage SP model considering various scenarios under the hydrogen supply chain. While a general stochastic programming approach fails to generate feasible solutions with volatile HSCN uncertainties, the proposed WSSAA framework provides feasible decisions considering multiple scenarios. The effectiveness of the proposed model applied with WSSAA is demonstrated through comparative experiments over the models used in existing studies. • Hydrogen Supply Chain Network (HSCN) framework considering multi-stage nonlinear processes • HSCN with Hydrogen demand and transport capacity-based uncertainties • HSCN decision making using newly designed multi-stage stochastic programming • Weighted Scenario Sample Average Approximation (WSSAA) for effective HSCN decisions [ABSTRACT FROM AUTHOR]

This article studies a dynamic advance scheduling problem where multi-type patients arrive randomly to book the future service of a diagnostic facility in a public healthcare setting. The demand for the diagnostic facility generally arises from multiple sources such as emergency patients, inpatients, and outpatients. It is challenging for public hospital managers to dynamically allocate their limited capacities to serve the incoming multi-type patients not only to achieve their heterogeneous waiting time targets in a cost-effective manner but also to maintain equity among multi-type patients. To address this problem, a finite-horizon Markov Decision Process (MDP) model is proposed to minimize the total expected costs under the constraints of maintaining equity. Because of the complex structure of the feasible region and the high-dimensional state space, the property characterization of optimal scheduling policy and the exact solution of the MDP are intractable. To solve the MDP model with high-dimensional state and action spaces, we reformulate the MDP as a multi-stage stochastic programming model and propose a modified Benders decomposition algorithm based on new dual integer cuts to solve the model. Based on real data from our collaborating hospital, we perform extensive numerical experiments to demonstrate that our proposed approach yields good performance. [ABSTRACT FROM AUTHOR]

We focus on the lot-sizing and scheduling problem with the additional considerations of machine eligibility, sequence-dependent setups, and uncertain demands. Multi-stage stochastic programming is proposed. We analyze the problem structure and suggest ways for modeling and solving large-scale stochastic integer programs. The analysis compares deterministic and stochastic model solutions to assess demand variance effects under the circumstances of increasing, fluctuating, and decreasing demands. The result shows that the expected cost performance of the stochastic programming model outperforms that of the deterministic model, in particular, when the demand is highly uncertain in the circumstance of an upward market trend. Our study can apply to the wafer fab manufacturing and other industries that heavily restricted by machine eligibility and demand uncertainties. [ABSTRACT FROM AUTHOR]

In hydro predominant systems, the long-term hydrothermal scheduling problem (LTHS) is formulated as a multistage stochastic programming model. A classical optimization technique to obtain an operational policy is the stochastic dual dynamic programming (SDDP), which employs a forward step, for generating trial state variables, and a backward step to construct Benders-like cuts. To assess the quality of the obtained policy (the cuts obtained over the iterations), a confidence interval is computed on the optimality gap. As the SDDP is a cutting-plane based method, it exhibits slow convergence in large-scale problems. To improve computational efficiency, we explore different regions in which the cuts are usually constructed by the classical algorithm. For that, the cuts in the forward step are translated using ideas related to the definition of Chebyshev centers of certain polyhedrons. Essentially, the cuts are lifted by a parameter that vanishes along with iterations without harming the convergence analysis. The proposed technique is assessed on an instance of the Brazilian LTHS problem with individualized monthly decisions per plant, indicating a higher policy quality in comparison with the classical approach, since it computes lower optimality gaps throughout the iterative process. [ABSTRACT FROM AUTHOR]

We address the dynamic design of supply chain networks in which the moments of demand distribution function are uncertain and facilities' availability is stochastic because of possible disruptions. To incorporate the existing stochasticity in our dynamic problem, we develop a multi‐stage stochastic program to specify the optimal location, capacity, inventory, and allocation decisions. Further, a data‐driven rolling horizon approach is developed to use observations of the random parameters in the stochastic optimization problem. In contrast to traditional stochastic programming approaches that are valid only for a limited number of scenarios, the rolling horizon approach makes the determined decisions by the stochastic program implementable in practice and evaluates them. The stochastic program is presented as a quadratic conic optimization, and to generate an efficient scenario tree, a forward scenario tree construction technique is employed. An extensive numerical study is carried out to investigate the applicability of the presented model and rolling horizon procedure, the efficiency of risk‐measurement policies, and the performance of the scenario tree construction technique. Several key practical and managerial insights related to the dynamic supply chain network design under uncertainty are gained based on the computational results. [ABSTRACT FROM AUTHOR]

Multi-stage stochastic linear programs (MSLPs) are notoriously hard to solve in general. Linear decision rules (LDRs) yield an approximation of an MSLP by restricting the decisions at each stage to be an affine function of the observed uncertain parameters. Finding an optimal LDR is a static optimization problem that provides an upper bound on the optimal value of the MSLP, and, under certain assumptions, can be formulated as an explicit linear program. Similarly, as proposed by Kuhn et al. (Math Program 130(1):177–209, 2011) a lower bound for an MSLP can be obtained by restricting decisions in the dual of the MSLP to follow an LDR. We propose a new approximation approach for MSLPs, two-stage LDRs. The idea is to require only the state variables in an MSLP to follow an LDR, which is sufficient to obtain an approximation of an MSLP that is a two-stage stochastic linear program (2SLP). We similarly propose to apply LDR only to a subset of the variables in the dual of the MSLP, which yields a 2SLP approximation of the dual that provides a lower bound on the optimal value of the MSLP. Although solving the corresponding 2SLP approximations exactly is intractable in general, we investigate how approximate solution approaches that have been developed for solving 2SLP can be applied to solve these approximation problems, and derive statistical upper and lower bounds on the optimal value of the MSLP. In addition to potentially yielding better policies and bounds, this approach requires many fewer assumptions than are required to obtain an explicit reformulation when using the standard static LDR approach. A computational study on two example problems demonstrates that using a two-stage LDR can yield significantly better primal policies and modestly better dual policies than using policies based on a static LDR. [ABSTRACT FROM AUTHOR]

A virtual energy storage (VES) model is proposed in this paper to accommodate renewable energy under a special market regulation. Such VESs can provide or consume electricity to the main power grid under the premise that the daily net electricity energy is balanced. Furthermore, a multi-stage distributionally robust optimization (MSDRO) model is set up in this paper to address the temporal uncertainties in the day-ahead economic dispatch model. Compared with the traditional two-stage distributionally robust optimization, the proposed multi-stage approach provides more flexibilities so that the decision variables can be adjusted at each time period, leading to a complex nested formulation. To efficiently solve the MSDRO model, a stochastic dual dynamic programming method is employed to decompose the original large-scale optimization model into several sub-problems in the stages, as two steps: forward pass and backward pass. In the forward pass, the expected cost-to-go function is approximated by piecewise-linear functions and then several samples are used to generate a lower bound; the backward pass will generate Benders’ cuts at each stage from the solution of the forward pass. The forward and backward passes are performed iteratively until the convergence is reached. Numerical results on an IEEE 118-bus system and a practical power system in China verify the proposed method. [ABSTRACT FROM AUTHOR]

Natural disasters (e.g., hurricanes) can cause widespread power outages within distribution networks and interrupted power supply to critical loads (e.g., grocery stores, hospitals, gas, fire, and police stations) that provide utility services. Microgrids are localized power grids that can incorporate solar/photovoltaic (PV) distributed generators (PV-DGs) and energy storage systems (ESSs) for stand-alone system operations independent of the main grid, known as the island mode. This study investigates a microgrid design problem using PV-DGs and ESSs when facing prolonged power outages in the main grid. We propose a multi-stage stochastic program that holistically considers the techno-economics of microgrid investment and daily operations by optimizing the reliability and resilience of the microgrid during a week-long power outage. The model is designed from a utility perspective that includes budget constraints for investment. Due to the large model size, we develop a nested L-shaped algorithm that solves the problem exactly and analyzes the microgrid’s reliability across different weather scenarios in the entire decision-making horizon. Results from a case study using real-world data show that an islanded utility-scale microgrid can effectively provide uninterrupted power supply to a network of 5 and 10 critical loads, covering 100% and 97% of the demand in all possible future scenarios, with potential investments of $8 million and $15 million, respectively.

Hydrogen fueling stations (HFSs) with onsite hydrogen production systems, which are usually composed of electrolyzers, hydrogen storage tanks and fuel cells, not only supply hydrogen for hydrogen-powered vehicles but also serve as a dispatchable technology that can bid in electricity markets. Except participating in energy market, joining in reserve market can compensate the cost in energy market and increase the total revenue of HFS. This paper proposes a multi-stage stochastic programming model to find the optimal offering strategy of the HFS in energy, reserve markets taking into account a series of uncertainties: day-ahead price, secondary reserve price, system imbalance price and hydrogen demand. Nonanticipativity constraints are employed to guarantee the decisions are made according to the realized uncertainty information up to the present stage. Compared with traditional stochastic programming model, the proposed model adequately considers the sequential bidding decisions with the gradual revealing of the uncertainty over time. Numerical experiments based on one case study indicate that the participation of reserve market greatly increase the revenue of HFS. In addition, the proposed multi-stage stochastic programming model is effective in characterizing the sequential decision. • This paper proposes a multi-stage stochastic programming model. • To find the optimal offering strategy of the HFS in energy, reserve markets. • A series of uncertainties such as day-ahead price, and hydrogen demand are considered. • Nonanticipativity constraints are employed to guarantee the sequential decisions. [ABSTRACT FROM AUTHOR]

We investigate a project scheduling problem in which cash flows are periodically variable, and we aim to maximize the net present value (NPV). For each activity, a set of cash flows is considered where each one pertains to a particular period. This setting is compatible with inflation rates that may well occur in some countries with unstable economic situations where the occurrence times of inflation are sometimes known in advance. In this case, the project can be scheduled more suitably to abate probable pitfalls. In this paper, we investigate the problem in two settings: deterministic and stochastic cash flows. In the deterministic case, in each period we consider the expected value of cash flows of each activity as a constant and develop an integer linear programming model in conjunction with a branch-and-bound algorithm to solve the problem optimally. Moreover, we develop a multi-stage stochastic programming (MSSP) model to formulate the stochastic version of the problem. Using a set of randomly generated test instances and extensive computational results, we analyze the performance of our developed solution approaches. In addition, we compare the deterministic and stochastic models and analyze the sensitivity of the most important parameters. [ABSTRACT FROM AUTHOR]

A copula-measure Me based interval multi-objective multi-stage stochastic chance-constrained programming (CMIMOMSP) model is proposed for water consumption optimization. It can conduct water allocation amid multiple users and multiple stages, and deal with the uncertainties presented as interval numbers, random fuzzy interval numbers, and stochastic variables. It improves upon multi-stage stochastic chance-constrained programming by introducing the multi-objective programming, and it can tradeoff the relationships amid economic benefit, full usage of water resources, and economic loss. It enhances the accuracy of copula function and conditional distribution function through proposing the interval functions. Besides, it can deal with the impact of the decision attitudes of managers on water allocation by formulating the function equation between water demand and the optimistic-pessimistic factor. The CMIMOMSP model is applied to a case study of the Heihe River Basin to verify its application. The results indicate that: (1) the optimistic-pessimistic factors have different degrees of positive influences on water allocation for industrial, domestic and ecological sectors; (2) the joint violated probability and optimistic-pessimistic factor have various range of impacts on agricultural water allocation; (3) tthe objective function values have different variation tendencies with the rise of joint violated probabilities and optimistic-pessimistic factors. Its robustness is enhanced by comparing it with the three single-objective programming models. The CMIMOMSP model can provide various water allocation schemes for managers with different risk attitudes in semi-arid and arid districts. [ABSTRACT FROM AUTHOR]

In this work, we study a stochastic single machine scheduling problem in which the features of learning effect on processing times, sequence-dependent setup times, and machine configuration selection are considered simultaneously. More precisely, the machine works under a set of configurations and requires stochastic sequence-dependent setup times to switch from one configuration to another. Also, the stochastic processing time of a job is a function of its position and the machine configuration. The objective is to find the sequence of jobs and choose a configuration to process each job to minimize the makespan. We first show that the proposed problem can be formulated through two-stage and multi-stage Stochastic Programming models, which are challenging from the computational point of view. Then, by looking at the problem as a multi-stage dynamic random decision process, a new deterministic approximation-based formulation is developed. The method first derives a mixed-integer non-linear model based on the concept of accessibility to all possible and available alternatives at each stage of the decision-making process. Then, to efficiently solve the problem, a new accessibility measure is defined to convert the model into the search of a shortest path throughout the stages. Extensive computational experiments are carried out on various sets of instances. We discuss and compare the results found by the resolution of plain stochastic models with those obtained by the deterministic approximation approach. Our approximation shows excellent performances both in terms of solution accuracy and computational time. [ABSTRACT FROM AUTHOR]

This paper proposes a multi-objective mathematical model for designing an inter-modal transportation network that includes a seaport and multiple dry-ports in both export-oriented and import-oriented modes under sustainability and resilience criteria. A multi-objective particle swarm optimization algorithm is tailored to solve the large-sized instances. The model and its solution method are validated through several numerical examples in different sizes whose results demonstrate that the development and design of such a seaport-dry port network is a critical component of the government's economic development while its development is a challenging issue requiring proper financing. • Presenting a novel seaport-dry port network design model. • Tailoring a multi-stage stochastic model to account for data uncertainty. • Devising a MOPSO algorithm to solve the large size instances. [ABSTRACT FROM AUTHOR]

• A primal decomposition algorithm is proposed. • The algorithm is applied in a berth allocation problem under uncertainty. • A multi-stage stochastic programming model is formulated. • A method for scenario reduction is designed for the algorithm. Berth planning is an important decision in port operations. The uncertainties in maritime transportation may result in uncertain ship arrival and service times at a port for every week of a planning horizon. In a realistic maritime transportation environment, the uncertain information on ship arrival and service times for a week become known only after a decision is made in the previous week. This study proposes a multi-stage stochastic integer programming (SIP) model for a tactical-level port berth planning problem under uncertainty, which tries to make fixed baseline berthing plans to fit shipping liners' preferred time slots and reduce their expected delay costs with actual ship arrival and service times for all the weeks of a planning horizon. We propose an original primal decomposition algorithm to solve the multi-stage SIP model. The proposed algorithm passes primal columns of subsequent-stage problems to the first-stage problem to approximate the subsequent-stage decision-making. This algorithm can be generalized to a variety of similarly structured multi-stage SIP models. Using actual berthing data from Xiamen port, we conduct experiments to validate the efficiency of our primal decomposition algorithm. We also conduct experiments to quantify the benefit of using stochastic programming to model the berth planning, the benefit of modelling the problem as a multi-stage program, the benefit of the scenario reduction method designed in this study, and the algorithmic scalability. The proposed multi-stage SIP model for berth planning as well as the primal decomposition algorithm could be potentially useful for port operators to improve operational efficiency of container terminals in uncertain environments. [ABSTRACT FROM AUTHOR]

This paper presents a multi-period active distribution network planning (ADNP) with distributed generation (DG). The objective of the proposed ADNP is to minimize the total planning cost, subject to both investment and operation constraints. The paper proposes a multi-stage stochastic optimization model to address DG uncertainties over several periods, in which the decisions are made sequentially by only using the present-stage information. A nested decomposition method is proposed which applies the stochastic dual dynamic integer programming (SDDIP) method to address computational intractabilities of the proposed ADNP approach. The presented numerical results and discussions on a 33-bus distribution system and a large-scale 906-bus system verify the effectiveness of the proposed ADNP method and its solution method. [ABSTRACT FROM AUTHOR]

To meet low latency and real-time monitoring demands of IoT-eHealth, fog computing is envisioned as a key technology to offer elastic computing resource at the edge of networks. In this context, eHealth devices can offload collected healthcare data or computational expensive tasks to a nearby fog server. However, the mobility of the eHealth devices may make the connection between them to fog servers uncertain, resulting in possible migration between fog servers. In order to evaluate the impact of this uncertainty on decision-making for offloading and resource allocation, we formulate the task offloading problem as a Multi-Stage Stochastic Programming (MSSP), with aim of minimizing the total latency of offloading to determine whether to offload or not, how much workload to offload, how much computing resource to allocate, as well as whether to migrate or not. Different from the previous MSSP based work focusing on the workload assignment only, the proposed MSSP examines joint decisions of offloading, resource allocation, and migration, advancing the understanding of the interactions among these decisions. Furthermore, to reduce the computational complexity of MSSP, we design an efficient sub-optimal offloading policy based on Sample Average Approximation, called SAA-MSSP. We conduct extensive simulation experiments to validate the effectiveness of SAA-MSSP. The results show that SAA-MSSP can converge to a near-optimal solution quickly. [ABSTRACT FROM AUTHOR]

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In this paper, we propose a goal-based investment model that is suitable for personalized wealth management. The model only requires a few intuitive inputs such as size of wealth, investment amount, and consumption goals from individual investors. In particular, a priority level can be assigned to each consumption goal and the model provides a holistic solution based on a sequential approach starting with the highest priority. This allows strict prioritization by maximizing the probability of achieving higher priority goals that are not affected by goals with lower priorities. Furthermore, the proposed model is formulated as a linear program that efficiently finds the optimal financial plan. With its simplicity, flexibility, and computational efficiency, the proposed goal-based investment model provides a new framework for automated investment management services. [ABSTRACT FROM AUTHOR]

Motivated by the challenges encountered in sawmill production planning, we study a multi-product, multi-period production planning problem with uncertainty in the quality of raw materials and consequently in processes yields, as well as uncertainty in products demands. As the demand and yield own different uncertain natures, they are modelled separately and then integrated. Demand uncertainty is considered as a dynamic stochastic data process during the planning horizon, which is modelled as a scenario tree. Each stage in the demand scenario tree corresponds to a cluster of time periods, for which the demand has a stationary behaviour. The uncertain yield is modelled as scenarios with stationary probability distributions during the planning horizon. Yield scenarios are then integrated in each node of the demand scenario tree, constituting a hybrid scenario tree. Based on the hybrid scenario tree for the uncertain yield and demand, a multi-stage stochastic programming (MSP) model is proposed which is full recourse for demand scenarios and simple recourse for yield scenarios. We conduct a case study with respect to a realistic scale sawmill. Numerical results indicate that the solution to the multi-stage stochastic model is far superior to the optimal solution to the mean-value deterministic and the two-stage stochastic models. [ABSTRACT FROM AUTHOR]

The main challenge with the penetration of variable renewable energy (VRE) in thermal-dominated systems has been the increase in the need for operating reserves, relying on dispatchable and flexible resources. In the case of hydro-dominated systems, the cost-effective flexibility provided by hydro-plants facilitates the penetration of VRE, but the compounded production variability of these resources challenges the integration of baseload gas-fired plants. The Brazilian power system illustrates this situation, in which the development of large associated gas fields economically depends on the operation of gas-fired plants. Given the current competitiveness of VRE, a natural question is the economic value and tradeoffs for expanding the system opting between baseload gas-fired generation and VRE in an already flexible hydropower system. This paper presents a methodology based on a multi-stage and stochastic capacity expansion model to estimate the optimal mix of baseload thermal power plants and VRE additions considering their contributions for security of supply, which includes peak, energy, and operating reserves, which are endogenously defined in a time-varying and sized in a dynamic way as well as adequacy constraints. The presented model calculates the optimal decision plan, allowing for the estimation of the economical tradeoffs between baseload gas and VRE supply considering their value for the required services to the system. This allows for a comparison between the integration costs of these technologies on the same basis, thus helping policymakers and system planners to better decide on the best way to integrate the gas resources in an electricity industry increasingly renewable. A case study based on a real industrial application is presented for the Brazilian power system.