Multi-stage Stochastic Programming Model Research Articles

Code and Data Repository for On the Value of Risk-Averse Multistage Stochastic Programming in Capacity Planning

This repository aims to demonstrate the value of risk-averse multistage stochastic programming models in capacity planning problems.

Multistage stochastic programming with a random number of stages: Applications in hurricane disaster relief logistics planning

We consider a logistics planning problem of prepositioning relief commodities in preparation for an impending hurricane landfall. We model the problem as a multi-period network flow problem where the objective is to minimize the total expected logistics cost of operating the network to meet the demand for relief commodities. We assume that the hurricane’s attributes evolve over time according to a Markov chain model, and the demand quantity at each demand point is calculated based on the hurricane’s attributes (intensity and location) at the terminal stage, which corresponds to the hurricane’s landfall. We introduce a fully adaptive multi-stage stochastic programming (MSP) model that allows the decision-maker to adapt their logistics decisions over time according to the evolution of the hurricane’s attributes. In addition, we develop a novel extension of the standard MSP model to address the challenge of having a random number of stages in the planning horizon due to the uncertain landfall time of the hurricane. We benchmark the performance of the adaptive decision policy given by the MSP models with alternative decision policies, including a static policy, a rolling-horizon policy, a wait-and-see policy, and a decision-tree-based policy, all based on two-stage stochastic programming models. Our numerical results and sensitivity analyses provide key insights into the value of MSP in the hurricane disaster relief logistics planning problem.

A multi-stage stochastic programming model for multi-mission selective maintenance optimization

This research introduces a novel selective maintenance model in the case of systems undergoing multiple consecutive missions. The model considers uncertainties related to future operating conditions during each mission. Within each maintenance break, various optional actions ranging from replacements which are perfect to imperfect and also minimal repairs can be chosen for individual components. Evaluating the probabilities of successful future mission accounts for uncertainties associated with component operational conditions. The selective maintenance problem is formulated as a nonlinear mixed-integer model for optimization, and computational challenges are addressed using the progressive hedging algorithm. Numerical examples validate the new proposed model and illustrate the benefits of the model by estimating a more realistic reliability level and lower maintenance cost.

An integrated blockchain-enabled multi-channel vaccine supply chain network under hybrid uncertainties

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.

Multistage stochastic programming for integrated network optimization in hurricane relief logistics and evacuation planning

AbstractIn this article, we study the integrated hurricane relief logistics and evacuation planning (IHRLEP) problem, integrating hurricane evacuation and relief item pre‐positioning operations that are typically treated separately. We propose a fully adaptive multistage stochastic programming (MSSP) model and solution approaches based on two‐stage stochastic programming (2SSP). Utilizing historical forecast errors modeled using the auto‐regressive model of order one, we generate hurricane scenarios and approximate the hurricane process as a Markov chain, and each Markovian state is characterized by the hurricane's location and intensity attributes. We conduct a comprehensive numerical experiment based on case studies motivated by Hurricane Florence and Hurricane Ian. Through the computational results, we demonstrate the value of fully adaptive policies given by the MSSP model over static ones given by the 2SSP model in terms of the out‐of‐sample performance. By conducting an extensive sensitivity analysis, we offer insights into how the value of fully adaptive policies varies in comparison to static ones with key problem parameters.

Multistage stochastic program for mitigating power system risks under wildfire disruptions

The frequency of wildfire disasters has surged fivefold in the past 50 years due to climate change. Preemptive de-energization is a potent strategy to mitigate wildfire risks but substantially impacts customers. We propose a multistage stochastic programming model for proactive de-energization planning, aiming to minimize economic loss while accomplishing a fair load delivery. We model wildfire disruptions as stochastic disruptions with varying timing and intensity, introduce a cutting-plane decomposition algorithm, and test our approach on the RTS-GLMC test case. Our model consistently offers a robust and fair de-energization plan that mitigates wildfire damage costs and minimizes load-shedding losses, particularly when pre-disruption restoration is considered.

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).

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.

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.

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.

Multistage stochastic fractionated intensity modulated radiation therapy planning

Intensity modulated radiation therapy (IMRT) is a widely used cancer treatment technique designed to target malignant cells. To enhance its effectiveness on tumors and reduce side effects, radiotherapy plans are usually divided into consecutive treatments, or fractions, that are delivered over multiple weeks. However, typical planning approaches have focused on finding the full sequence of radiation intensities prior to the treatment, or were restricted to a single treatment session. In this work, we investigate the fractionated IMRT planning problem that accounts for geometric motion-related uncertainty during treatment. We propose a novel multistage stochastic programming (MSP) modeling framework that incorporates the sequential decision-making nature of the problem and prevailing stochasticity in cancer treatment. We model the uncertainty in the form of a scenario tree following the multiple instance of geometry approximation, and solve the MSP model using stochastic dual dynamic programming considering a variety of risk measures. We conduct computational experiments on five test cases that are generated based on clinical data. Through extensive simulations, we show that our MSP model generates higher quality treatment plans compared to deterministic and two-stage program counterparts based on multiple performance measures. In particular, our model leads to a higher rate of tumor coverage and a lower rate of radiation exposure for healthy tissues. Accordingly, the proposed MSP framework can greatly contribute to the clinical practice in fractionated IMRT.

Dynamic container drayage booking and routing decision support approach for E-commerce platforms

E-commerce drayage platforms are relatively new to the container drayage industry. Their revenue management poses an enormous logistical challenge owing to the unpredictable demand coupled with complex operations. In this paper, we propose a novel dynamic container drayage booking and routing problem with uncertain demand that incorporates important features of this emerging business model from the perspective of an e-commerce drayage platform. We formulate a multi-stage stochastic programming model for problem followed by developing an effective two-phase heuristic method based on a multi-scenario approach to derive both service booking and fleet routing solutions. In the two-phase heuristic, a container drayage service booking sub-problem is solved at the first phase to determine whether each request should be accepted for service or rejected, as well as the time period for serving the accepted requests, while a fleet routing sub-problem is tackled at the second phase to generate work schedules for the tractors and trailers to transport the containers. We conduct comprehensive computational experiments to demonstrate the quality and value of the proposed two-phase heuristic, and reveal insights into the practical operations management of e-commerce drayage platforms.

Production planning in maintenance facilities under uncertain repair time

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.

A decomposition algorithm for multi‐item production planning with independent random demand

AbstractProduction planning in a multiproduct setting where the demands for different items are independent random variables that are also featured with a dynamic behavior over the planning horizon is a challenging task. With a particular focus on maintenance facilities, this study proposes a multistage stochastic programming (MSP) model for operations planning under independent random demand of faulty components in the modular‐structured devices (e.g., gas turbines) received for repair and overhaul services. A Lagrangian relaxation‐based decomposition heuristic is also developed to efficiently solve the problem for real‐size instances. This heuristic relies on decomposing the MSP model into submodels corresponding to component STs and coordinating them via a subgradient algorithm to obtain a high‐quality feasible solution. Our numerical experiments conducted on a range of problem instances endorse the significant value of incorporating demand uncertainty and the effectiveness of the proposed solution methodology in overcoming computational complexity.

Multistage stochastic programming for the closed‐loop supply chain planning with mobile modules under uncertainty

AbstractTraditional supply chains usually follow fixed facility designs which coincide with the strategic nature of supply chain management (SCM). However, as the global market turns more volatile, the concept of mobile modularization has been adopted by increasingly more industrial practitioners. In mobile modular networks, modular units can be installed or removed at a particular site to expand or reduce the capacity of a facility, or relocated to other sites to tackle market volatility. In this work, we formulate a mixed‐integer linear programming (MILP) model for the closed‐loop supply chain network planning with modular distribution and collection facilities. To further deal with uncertain customer demands and recovery rates, we extend our model to a multistage stochastic programming model and efficiently solve it using a tailored stochastic dynamic dual integer programming (SDDiP) with Magnanti‐Wong enhanced cuts. Computational experiments show that the added Magnanti‐Wong cuts in the proposed algorithm can effectively close the gap between upper and lower bounds, and the benefit of mobile modules is evident when the temporal and spatial variability of customer demand is high.

Medium-term multi-stage distributionally robust scheduling of hydro–wind–solar complementary systems in electricity markets considering multiple time-scale uncertainties

Joint trading of hydro–wind–solar complementary systems (HWSCSs) in the electricity market (EM) helps to reduce the imbalance cost and increase profits. However, multiple energy resources and market price uncertainties affect the trading strategies. Existing medium-term (MT) scheduling approaches assume that the probability distribution of the random variable is perfectly known. Short-term variations were also ignored, which led to revenue loss and trading risk. To address the above issues, this paper proposes an MT multi-stage distributionally robust optimization (MDRO) scheduling approach for a price-taking HWSCS in the EM. Firstly, hourly unit commitment (HUC) constraints are incorporated into the MT scheduling model to accurately capture short-term variations. A novel ambiguity set is designed based on the modified chi-square distance to address probability distribution uncertainties at two different time scales. Subsequently, an MDRO scheduling model is proposed to optimize the trading strategy. Finally, the proposed MDRO model is converted to a large-scale multi-stage integer programming problem based on linearization and reformation. The stochastic dual dynamic integer programming algorithm is modified to ensure computational tractability. Xiluodu-Xiangjiaba HWSCS, located in the Jinsha River in China, was selected as a case study. The results show that: 1) the MDRO model is more robust to distributional uncertainties than the multi-stage stochastic programming (MSSP) model. When the probability distribution of the random variable changes, the MDRO model yields a higher expected revenue (+2.43%) and a lower standard deviation (-60.8%) of revenue, which illustrates lower trading risk. 2) Compared with MSSP, deterministic, two-stage stochastic programming, and distributionally robust optimization models, the MDRO model exhibits the best out-of-sample performance in terms of the highest expected revenue and lowest trading risk. 3) Incorporating HUC constraints into the MDRO model helps to increase the total revenue (+3.53%) and energy generation (+3.31%) at the expense of increasing the computational burden.

A multi-stage stochastic programming approach for pre-positioning of relief supplies considering returns

A dynamic pre-positioning problem is proposed to efficiently respond to victims’ need for relief supplies under uncertain and dynamic demand in humanitarian relief. The problem is formulated as a multi-stage stochastic programming model that considers pre-positioning with the dynamic procurement and return decisions about relief supplies over a time horizon. To validate the advantages of dynamic pre-positioning, three additional pre-positioning strategies are presented: pre-positioning with one-time procurement and without returns, pre-positioning with one-time procurement and returns, and pre-positioning with dynamic procurement and without returns. Using data from real-world disasters in the United States in the Emergency Events Database, we present a numerical analysis to study the applicability of the proposed models. We develop a sample average approximation approach to solving the proposed model in large-scale cases. Our main contribution is that we integrate dynamic procurement and return strategies into pre-positioning to decrease both costs and shortage risks in uncertain and dynamic contexts. The results illustrate that dynamic pre-positioning outperforms the other three strategies in cost savings. It also indicates that a higher return price is particularly helpful for decreasing unmet demand. The proposed models can help relief agencies evaluate and choose the solutions that will have the greatest overall effectiveness in the context of different relief practices.

Modeling Multi-Objective Optimization with Updating Information on Humanitarian Response to Flood Disasters

Unpredictable natural disasters brought by extreme climate change compound difficulties and cause a variety of systemic risks. It is thus critical to provide possibilistic scheduling schemes that simultaneously involve emergency evacuation and relief allocation. But the existing literature seldom takes emergency evacuation and relief supplies as a joint consideration, nor do they explore the impact of an unpredictable flood disaster on the scheduling scheme. A multi-stage stochastic programming model with updating information is constructed in this study, which considers the uncertainty of supply and demand, road network, and multiple types of emergency reliefs and vehicles. In addition, a fuzzy algorithm based on the objective weighting of two-dimensional Euclidean distance is introduced, through moderating an effect analysis of the fuzzy number. Computational results show that humanitarian equity for allocating medical supplies in the fourth period under the medium and heavy flood is about 100%, which has the same as the value of daily and medical supplies within the first and third period in the heavy scenarios. Based on verifying the applicability and rationality of the model and method, the result also presents that the severity of the flood and the fairness of resources is not a simple cause-and-effect relationship, and the consideration of survivor is not the only factor for humanitarian rescue with multi-period. Specifically, paying more attention to a trade-off analysis between the survival probability, the timeliness, and the fairness of humanitarian service is essential. The work provides a reasonable scheme for updating information and responding to sudden natural disasters flexibly and efficiently.

A multi-stage stochastic programming model for the unit commitment of conventional and virtual power plants bidding in the day-ahead and ancillary services markets

As more uncontrollable renewable energy sources are present in the power generation portfolio, the need of more detailed and reliable tools for the optimal operation of energy systems has increased in the last years. This work presents a multi-stage stochastic Mixed Integer Linear Program with binary recourse for optimizing the day-ahead unit commitment of power plants and virtual power plants operating in the day-ahead and ancillary services markets. Scenarios reproduce the uncertainty of the ancillary services market requests, and production of photovoltaic panels. A novel decomposition algorithm is proposed to tackle the challenging multistage stochastic program. The methodology is tested on three types of large power plants: a natural gas-fired combined cycle, a combined heat and power combined cycle with thermal storage, and a virtual power plant integrating a combined cycle with battery and photovoltaic fields. Compared to the typical deterministic unit commitment approach, the proposed stochastic optimization approach allows to increase the revenues of the conventional power plant up to 13.58% and, for the combined heat and power and virtual power plant case, it allows finding a feasible and efficient operational scheduling.