Two-stage Stochastic Programming Model Research Articles

Design of Sustainable Renewable-Based Utility Plants in the Face of Uncertainty

This work presents the formulation of a two-stage stochastic mixed-integer linear programming (MILP) model to include uncertainty in the design of renewable-based utility plants. The model is based on a superstructure that integrates technologies to process biomass, waste, solar radiation, and wind and considers uncertainty in availability of the renewable resources and on the utility demands. The uncertain parameter space is calculated based on a monthly probability density function for each uncertain parameter and discretized into different levels. It is shown that as uncertainty is considered in the model formulation, design flexibility improves with respect to the deterministic-based designs, although the flexibility is achieved at the expense of higher underused facilities and therefore unused investment cost.

A scenario-based stochastic programming approach for the public charging station location problem

This paper presents an integrated framework for the optimal planning of public charging stations for plug-in electric vehicles (PEVs) in urban areas. The framework consists of two main components: (i) an out-of-home charging demand model based on an activity-based travel demand model, and (ii) a public charging station location-allocation model using a scenario-based stochastic programming (SP) approach. In order to capture the dynamic charging behaviour of PEV users, a chi-squared automatic interaction detector (CHAID)-based mixed effects decision tree is induced from multi-day activity diaries. Moreover, because the stochastic error of the micro-simulation approach brings about uncertainty, we adopted a two-stage stochastic mixed-integer programming (TSMIP) model, which measures uncertainty by means of a finite set of scenarios obtained from the derived decision rules underlying PEV charging. The proposed approach is demonstrated for the City of Eindhoven, The Netherlands, and benefits of the stochastic solution are discussed.

Multi-Objective optimization of the Hydrocarbon supply chain under price and demand uncertainty

ABSTRACT The energy sector is known for its enormous investments, despite the erratic behaviour of the oil market (e.g. changes in crude oil prices). Therefore, strategic and tactical planning of the Hydrocarbon supply chain (HCSC), considering market uncertainty, is a significant area of research. HCSC construction involves the integration of crude oil and natural gas supply chains (SCs). In this study, a stochastic multi-objective optimization model is developed for the tactical planning of HCSC. The model considers price and demand uncertainty by formulating a two-stage stochastic programming model. Financial objectives are considered in terms of cost minimization and revenue maximization, while a non-financial objective is considered in terms of depletion rate minimization (i.e. reserves sustainability maximization). The model assists the decision-maker in quantifying the amount of production required to meet demand under different scenarios. Furthermore, the proposed model assists in evaluating the trade-offs among alternatives. A real-world HCSC is used to elucidate the practicability of the model, and some managerial insights are derived by conducting a sensitivity analysis. For instance, production can be reduced during high demand periods to maintain enough reserves, and the excess demand can be satisfied from the outside market based on medium-term contracts.

A decomposition approach for the stochastic asset protection problem

The deterministic Asset Protection Problem (APP) involves deploying firefighting resources of various capabilities to service as many community assets as possible within time windows determined by an advancing wildfire. A common situation arising during these situations is a wind change. Forecasts of changes in wind velocity (i.e. direction and speed) are reasonably accurate but there is some uncertainty around the time of a wind change. The timing has implications for which areas, and hence which assets, will be impacted by the wildfire. This presents a difficult problem for an Incident Management Team (IMT) operating under severe time pressure as the wildfire sweeps across the landscape. In this study, we extend the spatial decomposition-based math-heuristic originally developed for the deterministic APP to handle large real life sized stochastic APPs in operational time. This is achieved by regarding the deterministic and stochastic components as a coupled system. A two-stage stochastic programming (TSSP) model is thus only used for a smaller sub-problem around the uncertainty. We found this approach outperformed other methods when tested on a new set of benchmarks. More importantly, the accuracy and solution times make the method suitable for operational purposes.

A risk-averse two-stage stochastic programming model for a joint multi-item capacitated line balancing and lot-sizing problem

In this paper, a comprehensive production planning problem under uncertain demand is investigated. The problem intertwines two NP-hard optimization problems: an assembly line balancing problem and a capacitated lot-sizing problem. The problem is modelled as a two-stage stochastic program assuming a risk-averse decision maker. Efficient solution procedures are proposed for tackling the problem. A case study related to mask production is presented. Several insights are provided stemming from the COVID-19 pandemic. Finally, the results of a series of computational tests are reported.

Resilient NdFeB magnet recycling under the impacts of COVID-19 pandemic: Stochastic programming and Benders decomposition

Neodymium-iron-boron (NdFeB) magnets are the most powerful magnets per unit volume sold in the commercial market. Despite the increasing demand for clean energy applications such as electric vehicles and wind turbines, disruptive events including the COVID-19 pandemic have caused significant uncertainties in the supply and demand for NdFeB magnets. Therefore, this study aims to alleviate the risk of supply shortage for NdFeB magnets and the containing critical materials, rare-earth elements (REEs), through the development of a resilient reverse supply chain and logistics network design. We develop scenarios to model the unique impact of the COVID-19 pandemic on the proposed business, incorporating both disruption intensity and recovery rate. We formulate a chance-constrained two-stage stochastic programming model to maximize the profit while guaranteeing the network resiliency against disruption risks. To solve the problem in large-scale instances, we develop an efficient Benders decomposition algorithm that reduces the computational time by 98.5% on average compared to the default CPLEX algorithm. When applied to the United States, the model suggests the optimal facility locations, processing capacities, inventory levels, and material flows for NdFeB magnet recyclers that could meet 99.7% of the demand. To the best of our knowledge, this study is the first to incorporate the impacts of the COVID-19 pandemic to design a resilient NdFeB magnet recycling supply chain and logistics network, leveraging risk-averse stochastic programming.

Effects of uncertainties on the capacity and operation of an integrated energy system

Uncertainty is a common and critical problem for planning the capacity and operation of integrated energy systems (IESs). This study evaluates the effects of uncertainties on the capacity and operation of an IES. To this aim, system planning and operation with uncertainties are optimized by a two-stage stochastic programming model and compared with a referencing deterministic case. Specifically, the uncertainties of photovoltaic (PV) generation and energy demand are investigated.Regarding system capacity, a larger energy storage capacity is needed to accommodate a higher uncertainty. The superimposed uncertainties have a higher effect on system capacity than the sum of the effect of each uncertainty. The uncertainty of energy demand has a higher impact than the uncertainty of PV generation.Regarding system operation, the increase in operation cost is smaller than the increase in investment cost and total cost. In addition, the average flexibility provided by the energy storage increases with uncertainty and uncertainties affect the change rate for power charging/discharging of the electric energy storage. Regarding the effect on the grid, the uncertainties increase not only the magnitude of ramping-rate, but also the frequency of power-dispatch.

An Optimization Model for Water Management under the Dual Constraints of Water Pollution and Water Scarcity in the Fenhe River Basin, North China

Sustainable watershed development suffers from severe challenges, such as water pollution and water scarcity. Based on an analysis of water quality and water utilization in the Fenhe River Basin, an inexact two-stage stochastic programming model with downside-risk aversion was built for optimal water resource allocations for the four primary water use sectors (industry, domestic use, agriculture, and the environment) in the Fenhe River Basin. The model aims to maximize the comprehensive watershed benefits, including water benefits, water costs, water treatment costs, and downside risks. The constraints are water quality, available water resources, and sectoral demands in different hydrological scenarios. The results show that pollutant emissions decrease as risk-aversion levels increase and show the opposite trend in the midstream and downstream areas. The increase in water resource allocation for agriculture and reduction in ecological water indicate that agriculture suffered the greatest water shortage and risk. Improving water recycling and coordinating the transferred water resources increases the comprehensive benefits and reduces sectoral risks. The model effectively manages rational water allocations under dual constraints and provides support for coordinating socio-economic development and environmental protection in the river basin.

Optimal Operation of Combined Energy and Water Systems for Community Resilience against Natural Disasters

One of the most critical challenges for modern power systems is to reliably supply electricity to its consumers during and in the aftermath of natural disasters. As our dependence on electrical power has increased over the years, long-term power outages can lead to devastating impacts on affected communities. Furthermore, power outages can halt the operation of water treatment plants, leading to shortages in clean water, which is essential during post-disaster recovery. One way to address this is to temporarily reconfigure power and water networks into localized networks, i.e., electric microgrids and water micro-nets, that utilize local resources to supply local demand independently of the main power grid and/or water network. Utilizing distributed energy resources such as wind and solar and treating wastewater locally for potable reuse can provide the operational flexibility for such systems to operate sustainably. However, due to uncertainties in both renewable energy generation and electric/water consumption, ensuring sustainable operation is a challenging task. In this paper, an optimal operational strategy is proposed for an islanded microgrid/micro-net, considering the stochastic nature of renewable energy resources, electric demand, and water demand. An energy storage system is modeled to address the uncertainty in power generation and demand, in conjunction with local water storage and wastewater treatment to accommodate variable water demands. A two-stage stochastic programming model is formulated and solved to determine an optimal operation strategy for the combined system.

Vehicle substitution in heterogeneous round-trip carsharing systems

Carsharing has become a viable alternative mode of transportation which not only contributes to a better environment and less traffic congestion, but is often also cheaper for its users. It is a challenging task for carsharing system operators to design an efficient system which meets user demand while at the same time limiting operational expenses. This problem becomes even more difficult if the provider offers a fleet of heterogeneous vehicles to users as this introduces the opportunity to substitute vehicle types. Substitution occurs when the requested vehicle type is not available and the user is instead assigned to a different, typically larger, vehicle. This paper investigates the main drivers of vehicle substitution in a round-trip carsharing system and how it affects profit. To account for uncertainty of demand, a two-stage stochastic programming model is proposed in which variables associated with user requests are considered random. A sample average approximation approach is used to find solutions within acceptable computation time. A computational study shows that the proposed method is able to closely approximate the true expected profit. Sensitivity analysis of different problem parameters demonstrates how the number of vehicles in use and the users’ spatial flexibility increase the number of substitutions. A detailed analysis of solutions shows how vehicle substitution always positively impacts the expected profit.

Stochastic mixed-integer programming for a spare parts inventory management problem

The German Armed Forces provide an operation contingent to support the North Atlantic Treaty Organization (NATO) Response Force (NRF). To fulfill a mission, the NRF operates a number of technical systems, mostly vehicles. Each system is composed of several parts which might fail over time, and it can only be used again in the mission if all broken parts are replaced. For short deployments (e.g., one month), the NRF troops bring with them a tightly constrained “warehouse” of spare parts. To ensure an optimal use of the space, we present a two-stage stochastic programming model where in the first stage spare parts are chosen, then failures occur at random, and in the second stage the parts are assigned to the broken systems. We carry out a scenario-based approach, where the failures are simulated by a Monte-Carlo approach. We demonstrate that the resulting mixed-integer linear program can be solved using standard numerical solvers. Using real-world input data provided by the German Armed Forces and simulated data, we analyze the sensitivity of the solutions with respect to the size of the warehouse, the service level, and the number of scenarios, and compare our approach with simpler, doctrine based warehousing strategies.

Stochastic stope design optimisation under grade uncertainty and dynamic development costs

ABSTRACT Stope design optimisation defines three-dimensional extraction volumes aiming to maximise cashflows, subject to geotechnical and operational constraints. Available stope layout methods are deterministic, failing to account for grade uncertainty and variability that affect stope locations and sizes, as well as value. A two-stage stochastic integer programming model for stope design optimisation is proposed, integrating grade uncertainty quantified through geostatistical simulations, level allocation, variable stope and pillar sizes for different geotechnical zones, and development costs. An application at an underground gold mine employing sublevel open stoping highlights the integration and management of grade uncertainty to define risk-resilient stope design.

An innovative approach to solve the carsharing demand-supply imbalance problem under demand uncertainty

This paper studies the demand-supply imbalance problem for one-way carsharing systems under a combination of pricing strategy, relocations and access trips considering stochastic demand. A novel concept of a virtual zone is utilized to capture vehicle relocation range and client walking or biking distance constraints in one-way carsharing systems. The vehicle imbalance problem is further addressed by combining a long-term pricing strategy and real-time vehicle relocations in a two-stage stochastic programming model. In the first stage, the tactical decisions including fleet size and trip price are optimized, while anticipating the operational costs from the second stage. The second stage optimizes operational decisions under uncertain demand including vehicle relocations conditional on the tactical decisions in stage one. The model aims to maximize the profit of a carsharing company considering the fleet costs calculated in stage one and the expected operational costs and revenue obtained in stage two. A dedicated gradient search algorithm is developed to solve the two-stage stochastic programming and results are compared to a genetic algorithm and an iterated local search algorithm. The proposed model and corresponding solution approach are applied to a large-scale network with 50 zones and over 1000 vehicles in Suzhou, China. The application allows us to attain additional operational insight. Results suggest that increased prices for high demand stations during peak hours reduce demand while maintaining profitability of the system. It is also found that the real-time vehicle relocations and flexibility of clients to pick up vehicles at farther stations can increase demand service rate by as much as 10%.

Improved Benders decomposition for stochastic yard template planning in container terminals

Recent trends in pursuing green ports have had positive effects on terminal operators constructing more eco-friendly decision schemes. A yard template in a container port not only determines the scheduling of the terminal handling equipment by container allocation, but also affects the carbon emissions from yard trucks caused by loading and unloading operations, due to the differing degree of yard road congestion. Moreover, the uncertainty regarding the number of transporting containers makes it challenging to construct a robust yard template for a container terminal. Motivated by research pointing to the need to alleviate the exhaust pollution generated by port handling equipment, this paper presents a two-stage stochastic programming model for yard template planning, with the aim of minimizing the maintenance and manpower costs of activating yard cranes for container handling and of reducing the carbon dioxide emitted from yard trucks. More specifically, the paper considers the impact of road congestion on carbon dioxide emissions from the perspective of the increase in transportation time caused by yard truck interruptions. The uncertainty regarding the number of transport containers is reflected by taking into account numerous scenarios in which different numbers of containers are loaded onto (discharged from) vessels that visit the port periodically. In addition, an improved Benders decomposition-based solution method is designed to solve the proposed model with large-scale problem instances. Several experiments are performed to validate the effectiveness of the model and the efficiency of the method. The numerical results demonstrate that yard template planning that considers road congestion can effectively reduce the carbon emissions of yard trucks in loading and unloading operations.

Sustainable–responsive winner determination for transportation service procurement auctions under accidental disruptions

This paper investigates a new winner determination problem with sustainable and responsive concerns under disruption risks for transportation service procurement auctions. In such an auction, a fourth party logistics (4PL) provider purchases transportation services from multiple third party logistics (3PL) providers. We develop a two-stage stochastic mixed-integer programming model that incorporates two features, including (i) a sustainable–responsive constraint to quantitatively describe the sustainability and responsiveness performance, and (ii) an effective hybrid mitigation strategy that combines the fortification policy with the temporary outsourcing service policy to mitigate disruptions. To determine winning 3PLs, a reduced scenario-based approximation approach is developed to solve the model and obtain a nearly optimal solution very quickly. Numerical experiments are conducted to illustrate the effectiveness and applicability of our model and method by using randomly generated instances as well as real data of a Chinese 4PL firm. Compared with existing studies, sensitivity analysis indicates that our proposed model can significantly reduce the 4PL’s expected cost and improve the long-term competitive advantages. Managerial implications are also drawn for 4PLs to run a sustainable–responsive logistics system under disruptions in a cost-effective way.

A novel humanitarian and private sector relief chain network design model for disaster response

The private sector is widely regarded as a useful element in the emergency response to disasters. Humanitarian organizations tend to cooperate with the private sector to deliver aids to affected areas and mitigate the complexity of disasters. In this paper, we propose an efficient relief chain consisting of a humanitarian organization (HO) and third-party logistics providers (3PLPs) to tackle the issue of decentralized planning and decision-making in humanitarian logistics. The relief chain is expected to benefit both the HO and the 3PLPs. To this end, we developed a novel bi-level two-stage multi-objective stochastic programming model. The robust equivalent of the proposed model is also developed to cope with uncertainty. A Benders decomposition algorithm is tailored to solve large-scale problems. An actual earthquake is investigated as case-study to test the model's real-world applicability. Finally, a number of managerial insights are inferred by analyzing the findings of this research.

Unlocking emerging impacts of carbon tax on integrated energy systems through supply and demand co-optimization

Integrated energy systems (IES) can help achieve greater energy efficiency, and then ultimately promote a climate-neutral economy by utilizing local renewable resources. Demand-side energy-saving measures can reduce operational costs associated with energy usage. Most existing IES models, however, focus on supply-side optimization, while the demand-side energy-saving potential and its impacts on whole-system performance are still not clear. The increasing carbon tax makes it even more important to understand the interactions between supply and demand sides to achieve a sustainable system with a minimal carbon charge. Hence, this study proposes a co-optimization model to simultaneously optimize the supply and demand sides of an IES considering the impact of the carbon tax. A selection tree is developed to describe various demand-side envelope upgrading technologies, and a binary tree is established by generating a set of supply-side scenarios with corresponding probabilities. Based on these results, an improved two-stage stochastic programming model is proposed. The robustness of the modeling results was further validated by a simulation–optimization-based uncertainty analysis addressing price uncertainties. A case study in Shanghai indicates that the proposed co-optimization model achieves more cost-efficient solutions than supply-side-only optimization considering carbon tax. Introducing carbon tax can reduce the installed capacity of fuel-based energy technologies by up to 24% and greatly accelerate the penetration of renewables. The increasing carbon tax also promotes the adoption of more advanced energy-saving technologies. Uncertainty analysis reveals acceptable robustness of the optimal demand-side scheme and supply-side configuration with a deviation of less than 5% and a coefficient of variation of 7%. Overall, the observations of the proposed model and case study provide valuable insights for IES design considering an emerging charge of carbon tax.

A Stochastic Mathematical Programming Approach to Resilient Supplier Selection and Order Allocation Problem: A Case Study in Iran Khodro Supply Chain

Suppliers as one of the main sources of vulnerability may lead to disruption and risk in supply chains. Thus, resilient supplier selection can lead to an increase in the resilience of the supply process, especially in automotive supply chains. The goal of this study is to select a set of resilient suppliers and optimal demand allocation in an automotive supply chain under risk. For this purpose, a bi-objective two-stage stochastic programming model is presented. In contrast to previous mathematical models, our model includes a new objective function to consider the supplier’s delivery performance as one of the criteria of resilient supplier selection and also the k-means clustering method is used to cluster and decrease the number of disruption scenarios. In the proposed model, due to the uncertainty of demand, chance-constrained programming approach has been utilized. The augmented Ɛ-constraint method is implemented to solve the presented model. Finally, sensitivity analysis has been done to determine the effect of parameter changes on the final results. The results of the research indicate that contingency planning can reduce the effect of disruption risks. The findings also show that the strategy of the supply chain regionalization is important in reducing the effects of environmental disruption.

Particle swarm optimization algorithm with time buffer insertion for robust berth scheduling

This paper investigates the robust berth allocation problem in container terminals. To handle the uncertainties in vessel arrivals, the problem is formulated as a scenario-based two-stage stochastic programming model. Furthermore, we introduce the time buffers to the model. We then develop an algorithm for time buffer insertion, which accommodates the adaptive search procedure for the time buffer into the Particle Swarm Optimization (PSO) algorithm. Different from the traditional PSO algorithm, a core operator is designed with a modified version to take the intelligent time buffer insertion approach. The results of the numerical experiments show that the proposed method consistently provides an improved performance in terms of solution quality, compared with the previous studies and the one with a traditional operator in the PSO algorithm.

Stochastic Programming for Liner Ship Routing and Scheduling under Uncertain Sea Ice Conditions

It is anticipated that in the foreseeable future the Northern Sea Route (NSR) will be able to serve commercial shipping as an alternative transportation shortcut between East Asia and Europe, especially in the summer season. The sailing time, however, is heavily subject to the variation of sea ice conditions along this route. Any participating shipping company must consider how to mitigate the ill effects on itinerary planning caused by sailing time and cost uncertainty. Finding a good trade-off between the benefit from a tight schedule and the risk caused by an unexpected delay is a key element in relevant routing and scheduling decisions, and may be beyond the reach of traditional deterministic planning models. With the aim of maximizing profit over all possible shipping environment scenarios, this article proposes a two-stage stochastic nonlinear integer programming model for liner ship routing and scheduling with uncertain shipping time and cost, the nonlinearity of which arises from the coexistence of schedule-sensitive shipping demand and uncertain arrival time variables in the objective function. The model is converted into an equivalent linear integer programming counterpart by introducing a set of nominal delay variables, and Benders decomposition is applied to solve the linearized problem. Numerical experiments and sensitivity analyses are conducted to validate the efficacy and effectiveness of the model, the results of which suggest several managerial insights that can be used to guide liner ship route and schedule planning under uncertain shipping conditions.