Multistage Stochastic Programming Research Articles

Decomposition methods for multi-horizon stochastic programming

Multi-horizon stochastic programming includes short-term and long-term uncertainty in investment planning problems more efficiently than traditional multi-stage stochastic programming. In this paper, we exploit the block separable structure of multi-horizon stochastic linear programming, and establish that it can be decomposed by Benders decomposition and Lagrangean decomposition. In addition, we propose parallel Lagrangean decomposition with primal reduction that, (1) solves the scenario subproblems in parallel, (2) reduces the primal problem by keeping one copy for each scenario group at each stage, and (3) solves the reduced primal problem in parallel. We apply the parallel Lagrangean decomposition with primal reduction, Lagrangean decomposition and Benders decomposition to solve a stochastic energy system investment planning problem. The computational results show that: (a) the Lagrangean type decomposition algorithms have better convergence at the first iterations to Benders decomposition, and (b) parallel Lagrangean decomposition with primal reduction is very efficient for solving multi-horizon stochastic programming problems. Based on the computational results, the choice of algorithms for multi-horizon stochastic programming is discussed.

Effective hydrogen supply chain management framework considering nonlinear multi-stage process uncertainties

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.

Liability-driven investment for pension funds: stochastic optimization with real assets

Using a multi-stage stochastic programming method, we suggest an optimal liability-driven investment (LDI) strategy for a closed defined-benefit pension fund including real assets. The objective is to jointly optimize contribution, funding ratio, and buyout cost, subject to a constraint on downside risk in terms of expected shortfall of assets relative to liabilities. Over a 10-year planning horizon, the optimal LDI strategy with a key-rate duration-matching bond portfolio outperforms the corresponding strategy with a duration-convexity matching bond portfolio as well as a strategy with an aggregate bond index-tracking portfolio. When real assets are introduced, the optimal LDI strategy includes significant investment in infrastructure and real estate, illiquidity notwithstanding. Nevertheless, delays in sales of real assets induced by illiquidity can increase downside risk.

Procurement portfolio management for resilient supply chains: value of information for risk averse decision making

AbstractIn this study, we investigate the replenishment decisions for firms that rely on multiple sources of supplies, including regular forward contracts, option contracts, and the spot market. Our research aims to tackle the challenges arising due to supplier disruptions and the volatility of spot market prices that are correlated with these disruptions. We develop and solve multi-stage stochastic programming models that incorporate demand and supplier disruption information updates, while considering both risk-neutral and risk averse (CVaR) objectives. These models assist organizations with varying risk attitudes in achieving maximum performance by optimally selecting a procurement portfolio based on the availability and quality of updated information. Through analytical solutions and extensive numerical studies, our findings offer novel insights to organizations and policymakers, empowering them to enhance their supply chain resilience during critical supply disruption situations. Therefore, this research has broader implications and aligns with the multiple sustainability objectives outlined in the Sustainable Development Goals (SDGs).

A Mutual Catastrophe Insurance Framework for Horizontal Collaboration in Prepositioning Strategic Reserves

Improving Disaster Preparedness Through Mutual Catastrophe Insurance In “A Mutual Catastrophe Insurance Framework for Horizontal Collaboration in Prepositioning Strategic Reserves,” H. Zbib, B. Balcik, M.-È. Rancourt, and G. Laporte present an innovative approach to collaborative disaster preparedness. The novel framework considers a risk-averse mutual insurer offering multiyear insurance contracts with coverage deductibles and limits to a portfolio of risk-averse policyholders. It is designed to foster horizontal collaboration among policyholders for joint disaster preparedness by effectively integrating operational and financial functions. The problem is modeled as a large-scale nonlinear multistage stochastic program and solved by using an effective Benders decomposition algorithm. The framework is validated with real data from 18 Caribbean countries focusing on hurricane preparedness. Given the predicted impacts of climate change, the proposed multiyear mutual catastrophe insurance framework promises to reshape global disaster preparedness and make a profound societal impact by providing a transparent disaster financing plan to protect vulnerable regions. The study’s findings stress the importance of long-term cooperation, prenegotiation of indemnification policies, and strategic setting of deductibles and limits by taking into account the correlation between policyholders.

Long-term operational planning for flexible residential buildings with seasonal storage and capacity-based grid tariffs

When performing operation of the energy system in a building, it is difficult to accurately represent the uncertain long-term objectives in the short-term window, and the correlation between long-term objectives. This work investigates a strategic modelling framework for representing long-term incentives during short-term operational planning for an energy system within flexible buildings. Multi-stage backwards Stochastic Dynamic Programming (SDP) is used to decompose the yearly operational problem into smaller stages, and create long-term cost curves that captures the cost of adjusting long-term price signals, which in this work considers seasonal thermal energy storage and recurring monthly capacity-based grid tariff costs. The proposed method is analyzed for a flexible realistic Norwegian building located in Southern Norway, where the influence both price signals have on long-term operational performance is analyzed. Results show that the framework strategically optimizes the long-term seasonal storage together with the recurring monthly grid tariff, complementing each other by using the seasonal storage to reduce peak costs during winter. With both seasonal storage and monthly demand charge, yearly operational costs are reduced by 4.3% compared to only accounting for the grid tariff. Additionally, the operational performance achieved only a 0.9% higher cost compared to yearly simulation with perfect information.

Approximation of multistage stochastic programming problems by smoothed quantization

AbstractWe present an approximation technique for solving multistage stochastic programming problems with an underlying Markov stochastic process. This process is approximated by a discrete skeleton process, which is consequently smoothed down by means of the original unconditional distribution. Approximated in this way, the problem is solvable by means of Markov Stochastic Dual Dynamic Programming. We state an upper bound for the nested distance between the exact process and its approximation and discuss its convergence in the one-dimensional case. We further propose an adjustment of the approximation, which guarantees that the approximate problem is bounded. Finally, we apply our technique to a real-life production-emission trading problem and demonstrate the performance of its approximation given the “true” distribution of the random parameters.

How a dedicated postdischarge unit can reduce hospital congestion and costs

AbstractDepending on the patient's condition, up to 60% of inpatients are discharged to post–acute care facilities (PACFs). These patients may experience several days of nonmedical inpatient stay until the hospital finds a facility that fits their needs, contributing to overcrowding in upstream units. This article studies the feasibility of creating a “postdischarge unit” (PDU) for medically ready‐for‐discharge patients who experience transfer delays, to improve access to inpatient beds. We use a multistage stochastic program, solved with a dual dynamic programming algorithm, to address the PDU size and capacity question. The random variable is the number of bed requests from upstream units (e.g., emergency department). Our numerical analysis, using data from a large hospital, shows that a PDU can reduce costs and significantly reduce the number of patients waiting for transfer to PACFs that are occupying inpatient beds, as long as the percentage of these patients in the hospital is more than 4%. Compared to current practice in our partner hospital, a PDU could increase access to inpatient beds by up to 13% and result in 2%–21% cost savings. Results show that PDU capacity in hospitals with a larger number of patients waiting for transfer is more sensitive to variation in PDU renovation and operational costs. In addition to using fewer medical staff, a PDU can improve discharge transitions to lower levels of care and more efficiently utilize social workers and physical therapists assisting these patients.

Which Is More Rewarding in Managing Sea-Level Rise and Hurricane Storm Surge Flooding: Mitigation or Response?

This study aims to extend the existing climate-change-induced flood mitigation research. We introduce an at-risk network to evaluate optimal cost–benefit strategies for creating dikes and levees to mitigate flood hazard over multiple years. Our proposed model includes the expected flood costs, estimated using possible climate-change-induced sea-level states throughout the planning horizon, and the investment costs for developing dikes and levees via land elevations across the at-risk network. Further, given the limitations on infrastructure investment, our model incorporates a budget constraint. The problem is modeled as a multistage stochastic program with recourse that minimizes overall expected costs over the planning horizon. Exploiting open-source and freely accessible data sets, the flood risk mitigation model elaborated here can be applied to most urban coastal situations due to its general nature. Using Boston as a case study, our proposed method resulted in a cost reduction of as much as 92.2%, with an average of 63.2%, compared to a “do nothing” strategy in a simulation-based experiment. Under a high sea-level rise scenario, the average cost savings observed by implementing the solution suggested by our model could be even 15% higher. This proposed approach offers decision-makers a tool to frequently assess the costs and risks faced by their cities enabling them to effectively mitigate the potential flooding risks.

Batch Learning SDDP for Long-Term Hydrothermal Planning

We consider the stochastic dual dynamic programming (SDDP) algorithm - a widely employed algorithm applied to multistage stochastic programming - and propose a variant using experience replay - a batch learning technique from reinforcement learning. To connect SDDP with reinforcement learning, we cast SDDP as a Q-learning algorithm and describe its application in both risk-neutral and risk-averse settings. We demonstrate the superiority of the algorithm over conventional SDDP by benchmarking it against PSR's SDDP software using a large-scale instance of the long-term planning problem of inter-connected hydropower plants in Colombia. We find that SDDP with batch learning is able to produce tighter optimality gaps in a shorter amount of time than conventional SDDP. We also find that batch learning improves the parallel efficiency of SDDP backward passes.

Multi-stage scenario-based stochastic programming for managing lot sizing and workforce scheduling at Vestel

Multi-stage scenario-based stochastic programming for managing lot sizing and workforce scheduling at Vestel

Combining deep reinforcement learning and multi-stage stochastic programming to address the supply chain inventory management problem

We introduce a novel heuristic designed to address the supply chain inventory management problem in the context of a two-echelon divergent supply chain. The proposed heuristic advances the current state-of-the-art by combining deep reinforcement learning with multi-stage stochastic programming. In particular, deep reinforcement learning is employed to determine the number of batches to produce, while multi-stage stochastic programming is applied to make shipping decisions. To support further research, we release a publicly available software environment that simulates a wide range of two-echelon divergent supply chain settings, allowing the manipulation of various parameter values, including those associated with seasonal demands. We then present a comprehensive set of numerical experiments considering constraints on production and warehouse capacities under fixed and variable logistic costs. The results demonstrate that the proposed heuristic significantly and consistently outperforms pure deep reinforcement learning algorithms in minimizing total costs. Moreover, it overcomes several inherent limitations of multi-stage stochastic programming models, thus underscoring its potential advantages in addressing complex supply chain scenarios.

Multistage stochastic dominance: an application to pension fund management

AbstractA pension fund manager typically decides the allocation of the pension fund assets taking into account a long-term sustainability goal. Many asset and liability management models, in the form of multistage stochastic programming problem, have been proposed to help the pension fund manager to define the optimal allocation given a multi-objective function. The recent literature proposes univariate stochastic dominance constraints to guarantee that the optimal strategy is able to stochastically dominate a benchmark portfolio. In this work we extend previous results (i) considering alternative types of multivariate stochastic dominance that appear more suitable in a multistage framework, (ii) proposing a way to measure the economic cost of introducing stochastic dominance constraints, (iii) proposing a sort of augmented stochastic dominance through a safety margin. Numerical results show the difference between the alternative ways to interpret and apply the multivariate stochastic dominance. These results are evaluated thanks to the proposed economic cost of the stochastic dominance constraints and either in presence or not of a safety margin.

Green and reliable medical device supply chain network design under deep dynamic uncertainty: A novel approach in the context of COVID-19 outbreak

Conditions governing industrial activities during and after global shocks with societal and economic transformations such as the COVID-19 pandemic have led to the loss of effectiveness of conventional approaches to dealing with uncertainties. The occurrence of sharp fluctuations in the essential parameters has left decision-makers in an unpredictable situation. Therefore, proactive efforts should be made to develop current approaches for adapting to new conditions. This paper establishes a strategic, tactical, and operational decision-making framework under the COVID-19 outbreak by developing a new uncertainty type called deep dynamic uncertainty. In the first step, a Mixed-Integer Linear Programming (MILP) model is proposed for the green and reliable closed-loop supply chain network design. The proposed model allows the decision-maker (DM) to manage and control co2emissions and e-waste generation. In the second step, a new three-step algorithm called Augmented Adjustable Column-Wise Robust Optimization (AACWRO) is first proposed. Then, by combining the proposed column-wise uncertainty with multi-stage stochastic programming (MSSP) approach, deep dynamic uncertainty is theorized for modeling the demand uncertainty under pandemic conditions. The model's performance under deep dynamic uncertainty has been carefully investigated based on the real ventilator and infusion pump supply chain network in Iran. The model under deep dynamic uncertainty, while maintaining tractability and adjustability, provides flexibility in entering data into the problem and significantly increases the coverage of modeling uncertainties. The results clearly demonstrate the efficiency of the proposed approach. The model under deep dynamic uncertainty at all levels of conservatism has on average 42.96% lower cost and 32% higher stability than the MSSP model.

A multistage distributionally robust optimization approach for generation dispatch with demand response under endogenous and exogenous uncertainties

AbstractDecision‐dependent (endogenous) uncertainties (DDUs), as a new type of uncertainties revealed recently, couple dispatch decisions with uncertainty parameters and thus render power system dispatch more challenging. However, most previous works handled various DDUs via stochastic programming (SP) or robust optimization (RO) in a two‐stage framework, which undoubtedly introduces the drawbacks of SP and RO, and cannot meet the nonanticipativity requirements in power scheduling. In this paper, a multistage distributionally robust optimization (DRO) method for generation dispatch with demand response (DR) is proposed considering the DDUs of deferrable loads and the decision‐independent (exogenous) uncertainties (DIUs) of wind power and regular loads. By analyzing the structure of decision‐dependency parameters, a novel data‐driven decision‐dependent ambiguity set is proposed, which provides a generic framework for formulating DDUs and DIUs simultaneously. Then a multistage DRO model with nested max‐min structure is developed to integrate the merits of DRO and nonanticipativity into generation dispatch. The proposed model is solved by tailored reformulation method and improved stochastic dual dynamic integer programming (SDDiP). Case studies illustrate the effectiveness of the proposed approach by comparing with the multistage SP, RO, and decision‐independent DRO methods.

How to operate ship fleets under uncertainty

Ships operated by a liner company are scattered around the world to transport goods. A liner company needs to adjust its shipping network every few months by repositioning its ships to respond to uncertain container shipping demand. Few studies investigate a liner company's multiperiod heterogeneous fleet deployment problem under uncertainty, considering fleet repositioning, ship chartering, demand fulfillment, cargo allocation, and adaptive fleet sizes. To this end, this study formulates a mixed‐integer linear programming model that captures all of these elements. This study also designs a Benders‐based branch‐and‐cut algorithm for this non‐deterministic polynomial‐time (NP)‐hard problem. Two types of acceleration strategies, including approximate upper bound tightening inequalities and Pareto‐optimal cuts, are applied to improve the performance of the algorithm. Extensive numerical experiments show that the proposed algorithm significantly outperforms CPLEX and its Benders decomposition framework in solving the model. We conduct an intensive analysis and find that multistage stochastic programming can lead to better solutions than two‐stage stochastic programming. We also find that 10% of the benefit provided by the multistage model over the two‐stage model is due to better fleet deployment decisions and that 90% of the benefit is due to better demand fulfillment and allocation decisions. By exploring three practical questions regarding driver analysis of liner company profitability, benefits analysis of adaptive fleet sizes, and the influence of the COVID‐19 pandemic on liner shipping, we show how liner companies can benefit from managerial insights obtained in this study.

Sustainability-based enterprise supply chain optimization and response under circular economy approach: agile, adaptive and coordinated

PurposeFrom a supply chain perspective, logistics firms collaborate with other supply chain members to extend their business scope. Investment in circular economy projects in the supply chain can not only broaden the scope of business but also increase the value of the entire supply chain. Third-party logistics companies are gradually participating in the construction and operation of many circular economy projects. How to coordinate multiple circular economy supply chain projects is at the core of its operation.Design/methodology/approachThis paper first analyzes some typical supply chain projects in China and summarizes the main features of these projects. Secondly, considering the benefits of the project and the stakes of each project, a multi-stage stochastic programming model is established. Finally, Cplex, nested decomposition, LocalSolver and other methods are adopted to simulate and analyze the model.FindingsThe final experimental results find that the importance of coordinating multiple circular economy supply chain projects to increase the value of the entire supply chain. The multi-stage stochastic programming model presented in this research can provide a useful tool for logistics enterprises and third-party logistics companies to optimize their investment decisions and maximize their profits in the context of a circular economy.Research limitations/implicationsThere are still some limitations to this study; for example, it is limited to the analysis of circular economy supply chain projects in China. The study focused on third-party logistics companies, and other enterprises in the circular economy supply chain were not considered. The research also assumed that the benefits of each circular economy project and the stakes of each project were known, which may not always be the case in real-world scenarios.Originality/valueThis manuscript found that investing in other circular economy projects in the supply chain can broaden the scope of business and increase the value of the entire supply chain. Third-party logistics companies are gradually participating in the construction and operation of many circular economy projects, such as recycling and repurposing initiatives. It highlights the importance of coordinating multiple circular economy supply chain projects to increase the value of the entire supply chain. The multi-stage stochastic programming model presented in this research can provide a useful tool for logistics enterprises and third-party logistics companies to optimize their investment decisions and maximize their profits in the context of a circular economy.

Optimal portfolio choice of couples with tax-deferred accounts and survival-contingent products

Financial products for retirement planning generally have complex taxation structures and death conditions. In particular, tax-deferred accounts (TDAs) can provide tax-sheltered wealth accumulation by deferring taxes, even with the same financial products. Additionally, various survival-contingent products (SCPs), such as annuity products and life insurance contracts, have different payouts for policyholders. In this study, considering both the TDA and SCPs, we formulate and solve a couple’s lifetime portfolio choice problem using a multistage stochastic programming model. Owing to its high-dimensional state space and lifelong planning periods, stochastic dual dynamic programming (SDDP) was used to solve this problem. We find some interesting results; when both the TDA and SCPs are available, the portfolio is less concentrated in annuity holdings than when the TDA is unavailable. Moreover, the couple ends their contribution to the TA earlier than when SCPs are unavailable.

Design of a Supply Chain-Based Production and Distribution System Based on Multi-Stage Stochastic Programming

Abstract 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.

A multi-stage stochastic programming approach for an inventory–routing problem considering life cycle

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.