Multi-stage Stochastic Programming Model Research Articles

A multi-stage stochastic programming model for sustainable closed-loop supply chain network design with financial decisions: A case study of plastic production and recycling supply chain

This paperproposes a multi-objective, multi-stage programming model to design a sustainable closed-loop supply chain network considering financial decisions. A multi-product, sustainable closed-loop plastic supply chain network design problem which encompasses economic, environmental and social objectives is modeled in a mathematical manner. The decisions to be made are concerned with location of facilities; the flow of products, loans to take and investments to make. Uncertainty issue is about demand of customers and investment's rate of return. The decision making model is formulated as a multi-objective, multi-stage mixed integer linear programming problem and is solved by implementing path formulation and augmented Ɛ-constraint methods. Computational analysis, is provided based on the subject company to determine the significance of the proposed model and the efficiency regarding integrating financial decisions with supply chain network design decisions.

Planning seasonal irrigation water allocation based on an interval multiobjective multi-stage stochastic programming approach

Water managers in arid and semi-arid areas must allocate limited irrigation water to different water use sectors considering the conflicting objectives, seasonal runoff inflow, and multiple uncertainties. To deal with these problems, an interval multiobjective multi-stage stochastic programming (IMMSP) model was proposed for finding reasonable water-storage scale and optimizing limited irrigation-water resources. Key factors in planning irrigation-water resources, such as random seasonal runoff, interval uncertainty in data collection, economic benefits, water leakage loss, and water deficit, were fully considered in the IMMSP model. Additionally, as an important indicator to describe the seasonal water supply capability of local water supply project, including reservoirs, agricultural ponds and etc., the water-storage scale, defined as the ratio of water storage capacity over total streamflow, is proposed to obtain the quantitative relationship between this indicator and objective of IMMSP. This study attempted to obtain the relationship between water-storage scale and objective of IMMSP as well as provide a reference of determining water-storage scale from the perspective of optimization. In addition, to solve the IMMSP model, a modified minimum deviation (MMD) method was proposed for dealing with uncertainties, making tradeoff among conflicting objectives, and reflecting the different importance of objectives. Both the IMMSP model and MMD method were applied to a real-world water-allocation problem in the middle reaches of the Heihe River basin for verifying its validity. The solutions generate a set of decision alternatives under different seasonal runoff scenarios and further guide local water managers identify the optimal management strategies and project construction scale of water conservancy. Moreover, a sound discussion of contribution to water-storage scale planning is made and the comparisons between IMMSP and each single-objective model (economic benefits, water leakage loss, and water deficit) in this study demonstrate that the results obtained by the proposed approach are more practical than a single objective with the same constraints. These results can not only effectively contribute to local irrigation water management and ecological restoration, but also provide more information to plan regional water-storage scale values.

Utilising demand response for distribution service restoration to achieve grid resiliency against natural disasters

The increased frequency of power outages due to natural disasters in recent years has highlighted the urgency of enhancing distribution grid resilience. The effective distribution service restoration (DSR) is an important measure for a resilient distribution grid. In this work, the authors demonstrate that DSR can be significantly improved by leveraging the flexibility provided by the inclusion of demand response (DR). The authors propose a framework for this by considering integrated control of household-level flexible appliances to vary the load demand at the distribution-grid level to improve DSR. The overall framework of the proposed system is modelled as a three-step method considering three optimization problems to (i) calculate feasible controllable aggregated load range for each bus, (ii) determine candidate buses to perform DR and their target load demand, and (iii) maintain the load level in each house through home energy management during the DSR, considering uncertainties in load and solar generation sequentially. The optimization problems are formulated as linear programming, mixed-integer linear programming, and multistage stochastic programming (solved using the stochastic dual dynamic programming) models. Case studies performed in the IEEE 123-node test feeder show improvements in resilience in terms of energy restored compared to the restoration process without DR.

A New Framework to Primal Bounding Multistage Stochastic Programs Under Endogenous and Exogenous Uncertainties

Optimization problems with both decision-dependent (endogenous) and decision-independent (exogenous) uncertainties are typical in the chemical process industry, especially in planning and scheduling. Stochastic programming is a framework for modelling optimization problems that involve uncertainty. Multistage stochastic programming (MSSP) is one approach for modelling such problems that consider decisions and recourse actions and that involve realization of uncertainties in multiple stages. However, MSSP models grow exponentially with increasing number of scenarios and time periods, and they quickly become computationally intractable for real-world problems. In this paper, we propose a general primal-bounding framework for large-scale MSSP models with both endogenous and exogenous uncertainties based on extending the concepts of expected value solution from MSSP under exogenous uncertainties. The proposed framework utilizes already known information and assumes the expected results for unrealized information to determine current state decisions. We applied the framework to solve instances of process-network-synthesis problem, which involves both uncertain process yields (endogenous uncertainty) and uncertain demand (exogenous uncertainty) with up to 1024 scenarios. The computational results reveal that proposed approach yielded feasible solutions within 22.3%, 13.4%, and 6.5% of the true solutions for the first, second and third instances, and obtained these solutions up to three order of magnitude faster than solving the original MSSP models.

Multi-Stage Stochastic Programming to Joint Economic Dispatch for Energy and Reserve With Uncertain Renewable Energy

To address the uncertain renewable energy in the day-ahead optimal dispatch of energy and reserve, a multi-stage stochastic programming model is established in this paper to minimize the expected total costs. The uncertainties over the multiple stages are characterized by a scenario tree and the optimal dispatch scheme is cast as a decision tree which guarantees the flexibility to decide the reasonable outputs of generation and the adequate reserves accounting for different realizations of renewable energy. Most importantly, to deal with the “Curse of Dimensionality” of stochastic programming, stochastic dual dynamic programming (SDDP) is employed, which decomposes the original problem into several sub-problems according to the stages. Specifically, the SDDP algorithm performs forward pass and backward pass repeatedly until the convergence criterion is satisfied. At each iteration, the original problem is approximated by creating a linear piecewise function. Besides, an improved convergence criterion is adopted to narrow the optimization gaps. The results on the IEEE 118-bus system and real-life provincial power grid show the effectiveness of the proposed model and method.

Incentive-Based Coordination Mechanism for Renewable and Conventional Energy Suppliers

This paper proposes an incentive-based coordination mechanism between a wind energy supplier (WES) and a conventional energy supplier (CES) to achieve a Pareto improvement. To comply with its day-ahead schedules and hedge against the intermittent wind energy generation, the WES is allowed to outsource a back up power capacity from the CES via making a bilateral contract. However, unanimous agreement cannot always be achieved since each party plays on its own interest. We employ the concept of swing option contracts to further encourage the suppliers to reach an agreement of the contract. On the one hand, the WES can leverage the uncertainty of wind output by covering possible energy shortage from the CES. On the other hand, the CES can optimally allocate its energy capacity by participating into the electricity market and offering capacity to backup the shortage of energy from the WES. The bidding problem for each supplier is formulated as a multi-stage stochastic programming model, with the objective of maximizing the expected profit while maintaining a low level of risk. Unlike the traditional two-stage approach, the proposed multi-stage model can effectively capture the impact of rebidding process in the real-time market. We incorporate conditional-value-at-risk as a risk measure to characterize the effect of risk perception of suppliers on their bidding decisions. Meanwhile, a game theory based approach is developed to obtain the contract items between the suppliers. Implementation results on real cases are provided to illustrate the effectiveness of our proposed framework.

Multistage Stochastic Unit Commitment Using Stochastic Dual Dynamic Integer Programming

Unit commitment (UC) is a key operational problem in power systems for the optimal schedule of daily generation commitment. Incorporating uncertainty in this already difficult mixed-integer optimization problem introduces significant computational challenges. Most existing stochastic UC models consider either a two-stage decision structure, where the commitment schedule for the entire planning horizon is decided before the uncertainty is realized, or a multistage stochastic programming model with relatively small scenario trees to ensure tractability. We propose a new type of decomposition algorithm, based on the recently proposed framework of stochastic dual dynamic integer programming (SDDiP), to solve the multistage stochastic unit commitment (MSUC) problem. We propose a variety of computational enhancements to SDDiP, and conduct systematic and extensive computational experiments to demonstrate that the proposed method is able to handle elaborate stochastic processes and can solve MSUCs with a huge number of scenarios that are impossible to handle by existing methods.

A multi-objective multi-stage stochastic model for project team formation under uncertainty in time requirements

Team formation is one of the key stages in project management. The cost associated with the individuals who form a team and the quality of the tasks completed by the team are two of the main concerns in team formation problems. In this study, we consider simultaneous optimization of cost and quality in a team formation problem. Because these problems in a project usually arise in multi-period planning with uncertain parameters, a multi-objective multi-stage stochastic programming (MOMSP) model is developed and presented. Having sufficient scenarios to realistically represent real-world environments often leads to high computational complexity. Thus, we have adopted scenario cluster decomposition methods in our modeling and developed a heuristic algorithm. It is shown that the model can be solved for problems with practical size. The presented model and its solution methodology can be applied to different types of projects. In this study, a project that involves an overhaul of a set of aircraft is presented as a case study in which the goals are to minimize staffing costs and maximize the reliability of the aircraft by staffing workforce with high competency.

A multi-stage stochastic programming model for relief distribution considering the state of road network

As an important aspect in disaster operations management, relief distribution has been challenged by lots of factors, such as unpredictable occurrence time, intensity and location of secondary disasters (e.g. aftershocks and landslides, which usually occur after an earthquake), and availability of vehicles. A multi-stage stochastic programming model is developed for disaster relief distribution with consideration of multiple types of vehicles, fluctuation of rental, and the state of road network. The state of road network is characterized using uncertain and dynamic road capacity. The scenario tree is employed to represent the uncertain and dynamic road capacity, and demonstrate the decision process of relief distribution. A progressive hedging algorithm (PHA) is proposed for solving the proposed model in large-scale size. Based on a real-world case of Yaan earthquake in China, numerical experiments are presented to study the applicability of the proposed model and demonstrate the effectiveness of the proposed PHA. Useful managerial insights are provided by conducting numerical analysis.

A multi-objective production planning problem with the consideration of time and cost in clinical trials

Under increasingly challenging circumstances, pharmaceutical companies try to reduce the overproduction of clinical drugs, which is commonly seen in the pharmaceutical industry. When the overproduction is simply reduced without an efficient coordination of the inventories in the supply chain, the stock-outs at clinical sites and clinical trial delay can hardly be avoided. In this study, we propose a multi-objective model to optimize the production quantity, where the clinical trial duration and the total production and operational costs are minimized. The problem is formulated as a multi-stage stochastic programming model to capture the dynamic inventory allocation process in the supply chains. Since this problem's solving time and required memory can increase significantly with the increase of the stage and scenario numbers, the progressive hedging algorithm is applied as the solution approach in this paper. In the numerical experiments, we study this algorithm's performance and compare the solving efficiency with the direct solution approach. In addition, we identify the optimal production quantity of clinical drugs and give a discussion about the tradeoffs between the clinical trial delay and total cost.

Profitable and Energy-Efficient Resource Optimization for Heterogeneous Cloud-Based Radio Access Networks

As network operators invest more and more in infrastructure to keep up with the ever-increasing traffic demand, it has become important for them to operate networks in a profitable manner. The resulting expansion of network infrastructure also increases power consumption, which has a negative impact on both the environment and revenue. In this regard, the cloud-based radio access network (C-RAN), which is a promising next-generation network architecture, has gained much attention as a solution. In addition, by utilizing the macro base stations for coverage, the resulting heterogeneous C-RAN (H-CRAN) can further help optimize the network while increasing the complexity of resource optimization. In this paper, we study an optimal resource allocation for C-RANs to maximize profit while minimizing power consumption considering the inherent network uncertainties. Moreover, by allowing network operators to share networking resources among themselves, we show that the service outage can be minimized without installing additional bandwidth or base stations. The proposed multi-stage stochastic programming model makes a robust optimal decision that effectively responds to uncertainties from users’ mobility and service demand while maximizing both profit and energy efficiency. The extensive evaluation and comparison results show that the proposed solution can maximize profit and energy efficiency while minimizing the service outage under network uncertainties.

Planning of Cascade Hydropower Stations with the Consideration of Long‐Term Operations under Uncertainties

In the location‐related planning of a hydropower system, the consideration of future operations under uncertainties can make the decisions sustainable and robust. Then, it is of great importance to develop an effective approach that deals with the long‐term stochasticity due to the long‐lasting effects of the location selections. Thus, we propose a multistage stochastic programming model to optimize the planning decisions of cascade hydropower stations and the long‐term stochastic operations in an integrated way. The first stage (i.e., the planning stage) in the model deals with the location and capacity decisions of the hydropower stations, while the subsequent stages implement the scheduling decisions under each stagewise stochastic scenario. To address the curse of dimensionality caused by the long‐term stochastic operations, we further propose a novel dimensionality reduction approach based on dual equilibrium to transform the multistage model into a tractable two‐stage stochastic program. The applicability of our approach is validated by a case study based on a basin of Yangtze River, China, and corresponding sensitivity analysis.

Stochastic modeling of fuel procurement for electricity generation with contractual terms and logistics constraints

This work addresses the procurement of fuel to supply an uncertain demand for electricity generation. Fuel purchase decisions are made with considerable advance with respect to reception. Changes in demand determine corrective decisions and associated costs, such as advance decisions on delay or cancellation of contracts in order to meet storage and logistics constraints. We present a multi-stage stochastic programming model for the problem. Uncertainty is modeled as a discrete event tree of scenarios. Three heuristic strategies, relax and fix, rolling horizon and scenario aggregation, are proposed and applied to three case studies. Results show that when delay or cancellation decisions are taken in advance of the reception period, the relax and fix strategy performs better in terms of quality of the solution and processing time. The methodology allows to model relevant aspects of the problem and proposes a robust procurement decision that hedges uncertainty and establishes corrective actions.

Dynamically optimizing the strategic plan of mining complexes under supply uncertainty

Mining complexes are comprised of multiple mines and mineral processing streams, each governed by internal (mineral deposit, operation) and external (commodity prices) uncertainties, and must be optimized jointly to manage technical risk and maximize economic value. This study presents a method that optimizes annual production scheduling of an open pit mining complex by developing a solution that provides a unique strategic mine plan that integrates feasible alternatives over investment decisions along the life of the asset. Accordingly, the long-term optimization is presented as a dynamic plan, which allows planning upfront for possible configuration transitions due to new capital investments, facilitating change. This method uses an adapted multistage stochastic programming model which expands upon the two-stage framework by performing multiple recourse stages that are solved iteratively, allowing feasible mine designs in a scenario-tree structure. In this model, dynamic investment decisions are made sequentially over the mine production schedule of related mines, based on new information that becomes available in each time period; these decision variables activate costs and effects over the model, letting the optimizer choose the capital investments to be considered at the mining and/or processing components of the mining complex. A copper open pit mining complex is used to test the proposed model, with options to invest in the truck and shovel fleet, and a secondary crusher to increase related capacities. Results show a substantial probability that the mine design should branch, presenting an increased expected net present value of over US$170M compared to the two-stage stochastic formulation.

A multistage risk-averse stochastic programming model for personal savings accrual: the evidence from Lithuania

In this paper we consider the problem of choosing the optimal pension fund in the second pillar of Lithuanian pension system by providing some guidelines to individuals with defined contribution pension plans. A multistage risk-averse stochastic optimization model is proposed that can be used to plan a long-term pension accrual under two different cases: minimum and maximum accumulation plans as possible options in the system. The investment strategy of personal savings is based on the optimal solutions over possible scenario realizations generated for a particular participant. The concept of the risk-averse decision-maker is implemented by choosing the conditional value at risk as the risk measure defined by a nested formulation that guarantees the time consistency in the multistage model. The paper focuses on three important decision-making moments corresponding to the duration of periods to be modelled. The first period is a short-term accumulation, while the second period is a long-term accumulation with possibly high deviation of objective function value. The third period is designed to implement the concept of target date fund in the second pillar pension scheme as the subsequent need to protect against potential losses at risky pension funds. The experimental findings of this research provide insights for individuals as decision-makers to select pension funds, as well as for policy-makers by revealing the vulnerability of pension system.

Strategic Planning of Steam Assisted Gravity Drainage Reservoir Development under Reservoir Production and Oil Price Uncertainty

During the lifecycle of an oilfield project, well development is a critical phase due to intensive investments required for capital cost. Determining whether it is economical to develop an oilfield or not and finding the best plan for field development are the main concerns in this field of study. Drainage area development and production planning are crucial in a steam assisted gravity drainage (SAGD) project of the oil sand industry. In this study, an optimization framework for planning the development of SAGD drainage areas under uncertainty is presented. The proposed framework includes the following two major elements. First, a mixed integer optimization model is developed to arrange the multiperiod development plan of the drainage areas with consideration of capital and steam allocation restrictions. Second, uncertainties in crude oil price and reservoir production are investigated based on a multistage stochastic programming model. The results demonstrate that the method can effectively generate an e...

A stochastic dynamic programming approach for delay management of a single train line

Railway delay management considers the question of whether a train should wait for a delayed feeder train. Several works in the literature analyze these so-called wait-depart decisions. The underlying models range from rules of thumb to complete network optimizations. Almost none of them account for uncertainties regarding future delays. In this paper, we present a multi-stage stochastic dynamic programming (SDP) model to make wait-depart decisions in the presence of uncertain future delays. The SDP approach explicitly accounts for potential recourse actions at later stations in a look-ahead manner when making the decision in the current stage. The objective is to minimize the total delay experienced by passengers at their final station by recursively solving Bellman equations. We focus on a single train line but consider the effects on direct feeder and connecting trains. In an extensive numerical study, we compare the solution quality and computational effort of the SDP to other optimization approaches and simple heuristic decision rules that are frequently used in delay management. The SDP approach outperforms the other approaches in almost every scenario with regard to solution quality in reasonable time and seems to be a promising starting point for stochastic dynamic delay management with interesting future research opportunities.

A Lagrangian relaxation approach for stochastic network capacity expansion with budget constraints

In this paper, we consider capacity expansion for network models subject to uncertainty and budget constraints. We use a scenario tree approach to handle the uncertainty and construct a multi-stage stochastic mixed-integer programming model for the network capacity expansion problem. We assume that permanent capacity and spot market capacity are available, which can be used permanently or temporarily by the decision maker respectively. By relaxing the budget constraints, we propose a heuristic Lagrangian relaxation method to solve the problem. Two algorithms are developed to find tight upper bounds for the Lagrangian relaxation procedure. The experimental results show superior performance of the proposed Lagrangian relaxation method compared with CPLEX.

Adaptation and approximate strategies for solving the lot-sizing and scheduling problem under multistage demand uncertainty

This work addresses the lot-sizing and scheduling problem under multistage demand uncertainty. A flexible production system is considered, with the possibility to adjust the size and the schedule of lots in every time period based on a rolling-horizon planning scheme. Computationally intractable multistage stochastic programming models are often employed on this problem. An adaptation strategy to the multistage setting for two-stage programming and robust optimization models is proposed. We also present an approximate heuristic strategy to address the problem more efficiently, relying on multistage stochastic programming and adjustable robust optimization. In order to evaluate each strategy and model proposed, a Monte Carlo simulation experiment under a rolling-horizon scheme is performed. Results show that the strategies are promising in solving large-scale problems: the approximate strategy based on adjustable robust optimization has, on average, 6.72% better performance and is 7.9 times faster than the deterministic model.

An Integrated Scenario Ensemble-Based Framework for Hurricane Evacuation Modeling: Part 1-Decision Support System.

This article introduces a new integrated scenario-based evacuation (ISE) framework to support hurricane evacuation decision making. It explicitly captures the dynamics, uncertainty, and human-natural system interactions that are fundamental to the challenge of hurricane evacuation, but have not been fully captured in previous formal evacuation models. The hazard is represented with an ensemble of probabilistic scenarios, population behavior with a dynamic decision model, and traffic with a dynamic user equilibrium model. The components are integrated in a multistage stochastic programming model that minimizes risk and travel times to provide a tree of evacuation order recommendations and an evaluation of the risk and travel time performance for that solution. The ISE framework recommendations offer an advance in the state of the art because they: (1) are based on an integrated hazard assessment (designed to ultimately include inland flooding), (2) explicitly balance the sometimes competing objectives of minimizing risk and minimizing travel time, (3) offer a well-hedged solution that is robust under the range of ways the hurricane might evolve, and (4) leverage the substantial value of increasing information (or decreasing degree of uncertainty) over the course of a hurricane event. A case study for Hurricane Isabel (2003) in eastern North Carolina is presented to demonstrate how the framework is applied, the type of results it can provide, and how it compares to available methods of a single scenario deterministic analysis and a two-stage stochastic program.