Two-stage Stochastic Programming Model Research Articles

Multi-facility aggregate production planning with prosumer microgrid: A two-stage stochastic program

The U.S. industries accounted for 37% of the wind turbine capacity installed in 2020 and corporate solar purchases have totaled $22 billion over the last 10 years. However, affordability and intermittency of renewable generation are still the main obstacles for firms to power their operations using wind and solar energy. Aggregate production planning models that incorporate onsite renewable energy (APPMs-RE) are the state-of-the-art to facilitate renewable adoption. A large number of APPMs are available in the literature, but renewables-based APPMs are still inadequately investigated. This paper proposes a novel APPM-RE because it optimizes wide-ranging RE decisions such as microgrid capacity, energy storage, prosumer energy transactions, and demand response along with production, machine usage, and workforce levels to minimize cost. The model is a two-stage stochastic program considering uncertainties in product demand, labor and machine capacity, and power generation. The research goals are to (1) investigate the feasibility of decarbonizing the production, transportation, and warehousing operations, (2) examine the system affordability under practical hourly load requirements, (3) assess the cost-benefit of a prosumer energy transaction mechanism (PETM), (4) identify the effect in cost and microgrid capacity of time-of-use tariff, and (5) demonstrate the suitability of a two-stage stochastic programming model. The numerical experiments relying on climate analytics of six U.S. cities represent a broad scope of climate conditions. The managerial insights derived from this study are: (1) renewable microgrids with PETM may attain net-zero carbon operations with an affordable levelized cost of energy, and (2) time-of-use stimulates more adoption of solar generation than the flat rate.

A risk-averse approach for joint contract selection and slot allocation in liner container shipping

Liner shipping companies need to satisfy the shipping demand from both the long-term contracts and the spot market. In general, the container shipping demand of the long-term contract has more stable volumes but lower freight rates compared with the spot market demand. It is thus a critical problem for the shipping companies how to provide discriminative shipping services to these two types of shipping demand through contract selection and container slot allocation to simultaneously increase the shipping profit and control its variations resulting from the demand variations. To handle this problem, a two-stage stochastic programming model is constructed in this paper which adopts the Conditional-Value-at-Risk (CVaR) to measure the profit variations. In the first stage, the shipping company selects the long-term contracts from a candidate contract set to sign with customers under the uncertainties in the show-up rates of containers in long-term contracts, and the volumes and the freight rates of the spot market demands. In the second stage, after all uncertainties have been realized, the shipping company determines the slot allocation for both the contract and spot demands. With the introduction of the CVaR, classical Benders decomposition cannot be directly applied to solve the risk-averse model. This paper thus develops a risk-averse Benders decomposition method tailored for the model. Numerical experiments are also conducted to verify the effectiveness of the solution method and provide some meaningful managerial insights.

Stochastic scheduling of chemotherapy appointments considering patient acuity levels

The uncertainty in infusion durations and non-homogeneous care level needs of patients are the critical factors that lead to difficulties in chemotherapy scheduling. We study the problem of scheduling patient appointments and assigning patients to nurses under uncertainty in infusion durations for a given day. We consider instantaneous nurse workload, represented in terms of total patient acuity levels, and chair availability while scheduling patients. We formulate a two-stage stochastic mixed-integer programming model with the objective of minimizing expected weighted sum of excess patient acuity, waiting time and nurse overtime. We propose a scenario bundling-based decomposition algorithm to find near-optimal schedules. We use data of a major university hospital to generate managerial insights related to the impact of acuity consideration, and number of nurses and chairs on the performance measures. We compare the schedules obtained by the algorithm with the baseline schedules and those found by applying several relevant scheduling heuristics. Finally, we assess the value of stochastic solution.

Stochastic RWA and Lightpath Rerouting in WDM Networks

In a telecommunication network, routing and wavelength assignment (RWA) is the problem of finding lightpaths for incoming connection requests. When facing a dynamic traffic, greedy assignment of lightpaths to incoming requests based on predefined deterministic policies leads to a fragmented network that cannot make use of its full capacity because of stranded bandwidth. At this point, service providers try to recover the capacity via a defragmentation process. We study this setting from two perspectives: (i) while granting the connection requests via the RWA problem and (ii) during the defragmentation process by lightpath rerouting. For both problems, we present the first two-stage stochastic integer programming model incorporating incoming request uncertainty to maximize the expected grade of service. We develop a decomposition-based solution approach, which uses various relaxations of the problem and a newly developed problem-specific cut family. Simulation of two-stage policies for a variety of instances in a rolling-horizon framework of 52 stages shows that our stochastic models provide high-quality solutions when compared with traditionally used deterministic ones. Specifically, the proposed provisioning policies yield improvements of up to 19% in overall grade of service and 20% in spectrum saving, while the stochastic lightpath rerouting policies grant up to 36% more requests, using up to just 4% more bandwidth spectrum.Summary of Contribution: For handling the intrinsic uncertainty of demand in the telecommunications industry, this paper proposes novel stochastic models and solution methodology for two fundamental problems in telecommunications at operational level: (i) routing and wavelength assignment (RWA) and (ii) lightpath rerouting problem. Despite the vast literature on the RWA problem, stochastic optimization has not been considered as a viable solution for resource allocation in optical networks. We propose two-stage stochastic programming models for both problems and design efficient decomposition-based solution methods that use various relaxations of the models and a new family of cutting planes. Our extensive and rigorous numerical experiments show the significant merit of incorporating uncertainty into decision making, as well as the effectiveness of the decomposition framework and our newly designed family of cuts in enhancing the solvability of both models. This work opens new avenues to explore where the powerful stochastic programming literature can be leveraged to make operational decisions in telecommunications problems, a field that currently relies mostly on deterministic and heuristic solution methods.

Optimization of twice-daily direct flyover data collection for satellite observations at uncertain locations

This study investigates a means to achieve twice-daily direct flyovers of a ground target whose location is unknown at the time of launch. The initial orbital parameters were chosen such that the expected cost of maneuvering to observe the ground target twice per day could be minimized. Previous work on this problem has only included one direct flyover daily and has left viewing a second pass during the day as a happenstance event. A two-stage stochastic programming model was developed and solved using the sample-average approximation (SAA) method along with the L-shaped method. As adding a second direct flyover daily results in a nonlinear model that includes an arccosine function, we proposed an approximation approach for the arccosine function as part of a quadratic program and evaluated the accuracy of our approach. Our results showed that for some scenarios, the maneuver costs were too high to allow for twice-daily direct flyovers, but in other scenarios, the cost was much more reasonable. However, by selecting initial orbital parameters that allowed for the potential of maneuvering for twice-daily direct flyovers, once the scenario had been realized, the mission manager could still decide whether the satellites should be maneuvered to achieve twice-daily observations.

A two-stage stochastic programming model for the sizing and location of DERs considering electric vehicles and demand response

This article addresses the optimal distributed energy resources (DER) allocation and sizing into an LV distribution network (DN) considering demand response (DR) and high penetration of plug-in electric vehicles (PEV) in a two-stage stochastic programming framework. The model includes (i) economic variables related to the capital cost of the DERs capacity installed with their respective operational costs and (ii) operational technical constraints based on the AC power flow equations. The formulation estimates the distributed generation (DG) capacity in terms of apparent power, i.e., the capacity found by the model relates the active and reactive power injected from DG to the DN simultaneously. A linearization and a modified version of the genetic algorithm (GA) solve the mix-integer non-linear problem (MINLP), and the backward algorithm is applied to reduce the number of scenarios. The model is tested in the IEEE 69-bus system under 18 different scenarios of 24 h to measure the DERs capacity installed variability. Finally, the results showed that (i) the set of scenarios selected are decisive for the optimal capacity installed affecting the investment and the Grid dependence, (ii) the PEVs affect the total DERs capacity installed and the operation costs, and (iii) the reactive power provided by the DERs might supply the total reactive power demand.

Integrated berth and yard space allocation under uncertainty

Improving container ports’ operational efficiency to reduce the delays at ports concerns global port operators significantly. Optimizing the tactical level allocation of quay side and yard side resources to vessel calls is a typical lever to mitigate the delays in operations. Another lever is by strategical level planning that re-adjust the preferred visiting time windows of vessel calls. It aims to absorb the delays at the scheduling stage rather than in the operations stage. This paper integrates the planning and operations at container ports to jointly optimize strategical level planning and tactical level berth and yard space allocation under uncertain vessel arrival times and uncertain numbers of loading/unloading containers. The problem is formulated as a two-stage stochastic integer programming model. To solve it, we develop an original decomposition algorithm that passes columns of second-stage problems to the first-stage problem to approximate the second-stage decision making. Numerical experiments are conducted to validate the effectiveness and efficiency of our proposed algorithm. Some managerial implications for port operators are also obtained.

Optimal location and operation of waste-to-energy plants when future waste composition is uncertain

In many countries, waste management is increasingly geared towards a circular economy, aiming for a sustainable society with less waste generation, fewer landfills, and a higher rate of recycling. Waste-to-Energy (WtE) plants, which convert waste into heat and energy, can contribute to the circular economy by utilizing types of waste that cannot be recycled. Due to the varying quality of sorting and socio-economic conditions in individual regions, the waste composition differs between regions and has an uncertain future development. Waste composition significantly affects the operation of WtE plants due to differences in energy potential. This paper supports strategic capacity planning for waste energy recovery by introducing a two-stage stochastic mixed-integer linear programming model that captures waste composition uncertainty through scenarios of possible future development. The results of the model provide insights into the economics of operation and identify important factors in the sustainability of the waste handling system. The model is demonstrated on an instance with six scenarios for waste management in the Czech Republic for the year 2030. The solution of the proposed model is to build 14 new WtE plants with a total capacity of 1970 kt in addition to the four existing plants with a capacity of 831 kt. The annual energy recovery capacity is expected to increase almost four times to satisfy EU directives that restrict waste landfilling.

Order assignment and scheduling under processing and distribution time uncertainty

In response to increasingly fierce competition and highly customized demands, many companies adopt a distributed production model but manage their orders in a centralized manner. Coordination between multiple factories requires unified information and resources to provide a close match between supply and demand. One of the crucial tasks is to solve the order assignment and scheduling problem with uncertainties introduced by unexpected changes in upstream supply, labor supply, and transportation capacity. Managing uncertainties in production and distribution is important, as they can significantly interrupt and delay the timely and constant supply of orders if not appropriately managed. We address an order assignment and scheduling problem with direct distribution under uncertainties in processing and distribution time. The aim is to achieve a minimum of weighted sum cost and timeliness, which involves the optimization of the order assignments to multi-factory and production scheduling for orders at each site. We first formulate the problem as a two-stage stochastic programming model. To manage a large scale of possible scenarios, we apply a sample average approximation (SAA) method to approximate the model. We propose a novel model with fewer binary variables and big-M constraints. An exact logic-based Benders decomposition (LBBD) method is developed to deal with practical-sized instances. Numerical results indicate the superiority of our new model and the LBBD method. Managerial implications are discussed to demonstrate its advantages and potential applicability in practice.

Goal programming approaches for coordinating flock collection in the poultry industry

We present deterministic and stochastic goal programming models for coordinating of flock collection activities in a poultry processing company. The aim is to develop weekly schedules that balance three goals: ensuring a steady workload in the processing facility, reducing processing defects due to weight differences, and fulfilling production targets of farmers, while satisfying logistical constraints. Two deterministic goal programming models are proposed: a weighted model that considers the collective interests of farmers, and a min-max model that prevents large deviations from the production target for any individual farmer. Furthermore, two-stage stochastic programming models are developed, in which forecasts of the average flock weights are uncertain. The proposed approaches are applied to a real case study in Nova Scotia (Canada). Numerical results show that, compared to the weighted models, the min-max models considerably reduce the maximum expected deviation from optimality without significantly increasing the gross deviation from production targets. Furthermore, the stochastic models led to substantial improvements over the deterministic ones, thus justifying the transition to a two-stage planning procedure. The proposed stochastic min-max model was also shown to outperform the current manual approach in terms of reducing the average weight spread between flocks collected on the same day.

Two-stage robust telemedicine assignment problem with uncertain service duration and no-show behaviours

The current pandemic of COVID-19 has caused significant strain on medical center resources, which are the main plac healthcare managers to make an effective assignment plan for the patients and telemedical doctors when providing telemedicine services. Motivated by this, we present the first comprehensive study of a two-stage robust telemedicine assignment problem when three different sources of uncertainty are incorporated, including uncertain service duration, no-show behaviours of both patients and telemedical doctors. From an algorithmic viewpoint, we propose an efficient nested column-and-constraint generation (C&CG) solution scheme that decomposes the model into an outer level problem and an inner level problem. Our results show that we can solve the problems of realistic sizes within a reasonable time (e.g., up to 100 patients, 10 telemedical doctors, and 200 scenarios within two hours). On the empirical side, we demonstrate how the hyper-parameters make a balance between cost management and the coverage level of the served patients in the presence of three different sources of uncertainty. Our comparison with a two-stage stochastic programming model implies that our model is not overly conservative and seems to provide a relatively cheaper modeling alternative that requires much less information support when hedging against three different sources of uncertainty under a worst-case situation.

Benders decomposition for internal truck renewal decision in green ports

ABSTRACT To effectively decrease emissions and pollution emitted by equipment in container ports, many ports began to update equipment using diesel fuels to achieve the aim of green ports. This paper studies an internal truck renewal problem in container ports and proposes a two-stage stochastic programming model to optimally determine internal truck composition optimization adjustment including three renewal modes, i.e. purchasing, retrofitting, and chartering. An exact solution procedure based on Benders decomposition accelerating by Pareto-optimal cuts is developed for the proposed model. Given the problem background of Shanghai Yangshan Deep Water Port, comprehensive computational experiments are carried out to verify the effectiveness of the proposed mathematical model and the performance of the solution approach. According to the numerical experiments, some suggestions for the management of renewing internal trucks in green ports are summarized at the end of the paper.

Stochastic hub location problems with Bernoulli demands

This paper introduces the hub location problem with Bernoulli demands (HLPBD). The problem is a variant of capacitated stochastic hub location problems in which demands for transportation service are assumed to be Bernoulli random variables. We address the problem under single and multiple allocation settings, and for each case, a two-stage stochastic programming model is presented. Further, assuming a homogeneous demand probability, a closed-form expression for the recourse function is presented, and deterministic equivalent formulations are developed for the original stochastic problems. To solve large instances, exact solution algorithms based on Benders decomposition and Lagrangian relaxation are proposed. An extensive set of computational experiments on three well-known data sets is conducted to test the efficiency of the proposed models and solution algorithms, and also to evaluate the effect of different input parameters on the optimal solutions.

Optimal design and evaluation of electrochemical CO2 reduction system with renewable energy generation using two-stage stochastic programming

Electrochemical CO2 reduction (ECO2R) is being considered as a potential solution to the problem of managing excess electricity from renewable energy generation. In this study, ECO2R system coupled with a renewable energy generation system is designed and evaluated for commercial applicability. Mathematical models of the various components of an ECO2R system leading to methanol as the product are developed and compared with experimental data. A two-stage stochastic programming model that minimizes the overall cost of the coupled system is formulated to determine the optimal capacity of the system under uncertain weather conditions. The optimal design is carried out for two regions with different weather patterns: Denver, USA and Munich, Germany. The result of the optimized design shows that the ECO2R system is economically feasible at a methanol price of 550 USD/ton. In addition, the two-stage stochastic programming approach proves to be an effective way to address uncertain weather conditions in the design.

A Stochastic Nash Equilibrium Problem for Medical Supply Competition

In this paper, we study the competition of healthcare institutions for medical supplies in emergencies caused by natural disasters. In particular, we develop a two-stage procurement planning model in a random environment. We consider a pre-event policy, in which each healthcare institution seeks to minimize the purchasing cost of medical items and the transportation time from the first stage, and a recourse decision process to optimize the expected overall costs and the penalty for the prior plan, in response to each disaster scenario. Thus, each institution deals with a two-stage stochastic programming model that takes into account the unmet demand at the first stage, and the consequent penalty. Then, the institutions simultaneously solve their own stochastic optimization problems and reach a stable state governed by the stochastic Nash equilibrium concept. Moreover, we formulate the problem as a variational inequality; both the discrete and the general probability distribution cases are described. We also present an alternative formulation using infinite-dimensional duality tools. Finally, we discuss some numerical illustrations applying the progressive hedging algorithm.

Risk regulation of water allocation in irrigation areas under changing water supply and demand conditions

The uncertainty of the hydrological environment and unbalanced water resource allocation result in a high risk of irrigation water shortages in regional agriculture, which seriously affects the sustainable development of agricultural systems. In this paper, we propose a risk regulation based modeling approach for the optimal allocation of agricultural water resources in a complex stochastic environment. The approach includes a conditional value-at-risk (CVaR) model, two-stage stochastic programming (TSP) model, two-dimensional joint distribution probability (JP) model, fractal criteria, and a multiple forms of chance-constrained programming (CCP) model. The model can weigh the contradiction between the intended target and associated penalties attributed to unknown hydrological events, measure the risk between system benefits and expected losses in agricultural water allocation at different confidence levels, and address the randomness in the objective function and constraints (including the left end term, right end term, and left and right end terms). To verify the applicability of the method, it is applied to the Jinxi Irrigation District in China to optimize the allocation and risk regulation of limited water resources under the variable runoff conditions of the Songhua River and crop water demands in the irrigation area. By adjusting parameters such as risk preference and probability of violation, the risk of water shortages in the irrigation area can be regulated, and the multidimensional impacts of different water allocation schemes on agricultural economic benefits, social benefits, ecology and environment can be determined. The case study reveals that the CTSP-CCJP method is sensitive, applicable to complex and uncertain environments and important for the efficient use of agricultural water resources and risk reduction.

A two-stage stochastic programming model for selective maintenance optimization

The problem of selective maintenance exists in many multicomponent systems carrying out an alternating sequence of missions, with scheduled breaks where only a limited number of components can be maintained due to time limit. In this paper, a stochastic programming approach is proposed for determining an optimum maintenance plan to minimize maintenance costs and expected failure costs, while maximizing the probability of successful accomplishment of the next mission under uncertainties in future operating conditions. Traditionally, future operating conditions that affect failure time distribution when calculating reliability in selective maintenance models were assumed as deterministic. In this study, future operating conditions are assumed to be uncertain. The system is subject to several uncertain condition scenarios of exposure, conditional, usage, stress, etc. Each scenario is modeled with its associated occurrence probability. The presented model is a two-stage stochastic mixed-integer nonlinear programming model with fixed recourse, where the first stage is associated with maintenance decisions made before uncertainties are revealed, and the second stage is modeled as a recourse function which is related to the occurrence probability of system failure. A numerical example of a series-parallel system is used to demonstrate the effectiveness of the suggested model.

How should water resources be allocated for shale gas development? An exploratory study in China

Water scarcity has emerged as one of the most important global challenges of the twenty-first century. With rising demand for energy, and water being a critical input in energy production, the availability of water resources has put energy sustainable production under growing strain. While unconventional natural gas (especially shale gas) is seen as an important bridge for promoting the transition of energy system from high to low carbon, water availability is a significant constraint on the development of energy resources owing to the massive quantity of water used by the hydraulic fracturing. Against this background, our study aims to optimize the allocation of regionally scarce water resources for fostering integrated economic, social, and environmental growth in shale gas development plays. In light of the uncertainty inherent in the water supply management system for shale gas development, this work employed the Interval Two-stage Stochastic Programming (ITSP) to establish an optimal allocation model for water resources between wells jointly dispatched by surface water, underground water and reused water. The model predicted water scarcity, optimal water allocation, and the total benefit of the shale gas development water supply system under various scenarios. Furthermore, when compared to the Two-stage Stochastic Programming (TSP) model results, it was found that the ITSP model's interval value may present decision makers with more ideas and options than the TSP model. In addition, since the ITSP model is oblivious to the system risk issue, it incorporated robust optimization into the original ITSP model to build the Interval Two-stage Robust Stochastic Programming (ITRSP) model. Our findings were expressed as intervals that more accurately represent the actual optimal allocation of water resources, which also provided a broader decision-making space for decision makers in managing shale gas development water supply management schemes.

A two-stage stochastic programming model of coordinated electric bus charging scheduling for a hybrid charging scheme

This paper proposes a coordinated charging scheduling approach for battery electric buses (BEBs) in a hybrid charging scheme, i.e., both plug-in fast charging and battery-swapping charging modes are incorporated in a single charging station. To accommodate the uncertain battery energy consumption during bus operation, a two-stage stochastic program is formulated, where the first stage decision determines the battery inventory level of each station and the second stage determines the charging mode and designs when, where, and how long each bus should be charged. Future uncertainties associated with energy consumption are captured by a set of possible discrete scenarios from historical data. A progressive hedging algorithm is developed to decompose the two-stage stochastic program into sub-problems. A case study is conducted to verify the proposed models and solution algorithms.

An optimization approach for disaster relief network design under uncertainty and disruption with sustainability considerations

Human-made, natural, and unexpected disasters always cause human and financial losses to communities. Disaster management is a framework with proven performance to reduce the damage caused by disaster and supply chain disruptions. Transferring the injured people from affected areas to hospitals at the minimum possible time is a crucial goal in times of disaster. This paper develops a two-stage stochastic programming model to transport the injured people from affected areas to hospitals in the incidence of multiple disruptions at transportation links and facilities under uncertainties. Herein, economic, social, and environmental aspects of sustainability are considered, while simultaneous disruptions are managed to minimize the adverse impacts of the disasters. We aim to determine optimal locations to establish transfer points and flows between the relief network nodes with sustainability considerations. Ultimately, a case study in District 12 of Tehran, Iran is conducted to ensure the proposed model’s validity and performance. Various sensitivity analyses are also implemented to ensure the model’s effectiveness. The results indicate that disruptions in facilities and transportation links lead to increased relief time, hence has the most significant negative impact on relief operations.