Two-stage Stochastic Programming Approach Research Articles

Application of Stochastic Programming in Agricultural and Newsvendor Problems and It's Application in Real Life

The study of making the best decision under risk management in a variety of areas of our lives is known as Stochastic Programming. We will go through two-stage Stochastic Linear Programming approaches for a variety of real-world choice issues, as well as how to solve them. We will achieve this by constructing stochastic linear programming models based on real-world situations like the well-known Farmer's situation and News Vendors problems. The influence of pricing, Stochastic Integer Linear Programming problem, second stage Stochastic Integer Linear Programming problem, first stage Stochastic Binary Linear Programming problem, risk aversion problem, and continuous function for random variables based on two-stage SLP with the aid of Farmer's problem will all be examined. We will address the Newsvendor’s problem with Deterministic Equivalent Stochastic Linear Programming, an extension of Deterministic Stochastic Linear Programming for risk aversion with a high number of decision variables and restrictions, utilizing the two-stage Stochastic Linear Programming approach once more. Hand calculation is a challenging way to acquire the solution to the problems. As a result, we will use the programming language AMPL to design computer solutions for tackling both farmer and newsvendor difficulties. We will also utilize MATLAB to create graphs for the farmer's problem's continuous function. Dhaka Univ. J. Sci. 72(1): 30-45, 2024 (January)

Stochastic Modelling Frameworks for Dragon Fruit Supply Chains in Vietnam under Uncertain Factors

Managing uncertainties and risks is always a difficult but fascinating task in fresh fruit supply chains, especially when dealing with the strategy for the production and conveyance of fresh fruit in Vietnam. Following the COVID-19 outbreak, the confluence of economic recession and persistent adverse weather conditions has exacerbated challenges faced by dragon fruit cultivators. This research investigates a two-stage stochastic programming (TSSP) approach which is developed and served as a valuable tool for analyzing uncertainties, optimizing operations, and managing risks in the fresh fruit industry, ultimately contributing to the sustainability and resilience of supply chains in the agricultural sector. A prototype is provided to illustrate the complex and dynamic nature of dragon fruit cultivation and consumption in Vietnam. Data on the selling prices of dragon fruit were collected from several sources between 2013 and 2022 in Binh Thuan Province, Vietnam. The results were obtained from the model by using three different approaches in order of their versatility and efficacy: (1) Scenario tree generation; (2) Sample average approximation; (3) Chance-constrained programming.

A nationwide multi-location multi-resource stochastic programming based energy planning framework

With the escalating global energy demands, nations face greater urgency in diversifying their electricity market portfolio with various energy sources. The efficient utilization of these fuel resources necessitates the integrated optimization of the entire energy portfolio. However, the inherent uncertainty of variable renewable energy generators renders traditional deterministic planning models ineffective. Conversely, expansive stochastic models that capture all the complexities of the power generation problem are often computationally intractable. This paper introduces a scalable energy planning framework tailored for policymakers, which is adaptable to diverse geopolitical contexts, including individual countries, regions, or coalitions formed through energy trade agreements. A two-stage stochastic programming approach, including a scenario-based Benders decomposition method, is employed to achieve computational efficiency. Scenarios for representing parameter uncertainty are developed using a k-means clustering algorithm. An illustrative case study is presented to show the practical application of the proposed framework and its ability to inform policymaking by identifying actionable cost-reduction strategies. The findings underscore the importance of promoting resource coordination and demonstrate the impact of increasing interchange and renewable energy capacity on overall gains. Notably, the study shows the superiority of stochastic optimization over deterministic models in energy planning.

Optimal Sizing of Hybrid Renewable Energy System Using Two-Stage Stochastic Programming

Stochastic programming has become increasingly vital in energy applications, especially in the context of the growing need for renewable energy solutions. This paper presents a significant advancement in this field by introducing an efficient and robust algorithm for optimally sizing hybrid renewable energy systems. Utilizing a two-stage stochastic programming approach, the proposed algorithm addresses the challenges posed by the unpredictability of renewable energy sources. The proposed solution leverages the three-block alternating direction method of multipliers (ADMM), a cutting-edge technique that facilitates parallel computation and enhances computational efficiency. The distinctiveness of this method lies in its ability to solve complex stochastic optimization problems without compromising the mathematical integrity of the model. This is achieved by applying first-order optimality conditions, ensuring both robustness and efficacy. To demonstrate the practical applicability and superiority of the algorithm, a case study was conducted in a rural area of South Africa. The proposed algorithm was applied to design an optimal hybrid renewable energy system, and its performance was compared against traditional methods such as progressive hedging and Monte Carlo techniques. Results affirm the superiority of the approach, saving approximately 8.16% capital cost when compared to progressive hedging. In addition, the proposed algorithm outperforms the Monte Carlo method both in terms of CPU time and the number of cost function evaluations.

A multi-objective two-stage stochastic programming approach for wind farms and demand side provider decisions using the weighted metric method

The rapid expansion of wind-based renewable energy sources highlights the critical need to enhance reserve capacity within power systems. This research aims to develop an effective model for harnessing the combined potential of flexible demand-side resources, acting as demand-side providers (DSPs), to address the inherent unpredictability of wind power. The DSP framework comprises three demand response aggregators: the capacity market program (CAP), energy storage systems (ESSs), and plug-in electric vehicle parking lots (PEVs PLs). To achieve this, a multi-objective two-stage stochastic programming model is introduced. It not only minimizes operational costs for system operators but also maximizes returns for demand-side resource owners supervised by DSPs in the energy and reserve market environment. The problem is tackled using a mixed-integer linear programming (MILP) approach employing a weighted metric methodology. Within this framework, the Weibull probability distribution function (WPDF) is utilized to model wind power uncertainty, while the truncated Gaussian distribution function (GDF) encapsulates the uncertainty regarding electric vehicle owner behavior. The results demonstrate that employing this method for simultaneous programming of energy and reserve not only reduces system operator costs but also enhances profits for resource owners.

A Two-Stage Stochastic Programming Approach for the Design of Renewable Ammonia Supply Chain Networks

This work considers the incorporation of renewable ammonia manufacturing sites into existing ammonia supply chain networks while accounting for ammonia price uncertainty from existing producers. We propose a two-stage stochastic programming approach to determine the optimal investment decisions such that the ammonia demand is satisfied and the net present cost is minimized. We apply the proposed approach to a case study considering deploying in-state renewable ammonia manufacturing in Minnesota’s supply chain network. We find that accounting for price uncertainty leads to supply chains with more ammonia demand met via renewable production and thus lower costs from importing ammonia from existing producers. These results show that the in-state renewable production of ammonia can act as a hedge against the volatility of the conventional ammonia market.

Two-stage stochastic programming approach for fleet sizing and allocating rail wagon under uncertain demand

In this paper, a scenario-based two-stage stochastic programming model has been developed to determine the optimal number of wagons and allocation of full and empty wagons during different time periods to optimize profit. In allocating wagons, when the wagon arrives at a station, it is not possible to reallocate the wagon at the moment of arrival, this issue has not been addressed before. In addition, the possibility of renting wagons to/from other companies is concerned in the railway planning. These two actions have not been addressed before. In this paper, firstly, the research problem is formulated in the form of an integer programming model with deterministic data. Due to the inherent uncertainty in the transportation demands of the real case study, the deterministic model has been transformed into a scenario-based two-stage stochastic programming model with recourse. Then the L-shaped method was proposed to solve the stochastic model and by using it for 34 instances, its better efficiency than CPLEX software was shown. The stochastic model was solved using the L-shaped method and the results show that the studied company should rent 749 wagons to meet the demands, and on average, the stochastic model is expected to have about 1.1% added value in the company's profit compared to the deterministic model.

An Optimal Wind Farm Operation Strategy for the Provision of Frequency Containment Reserve Incorporating Active Wake Control

This study proposes a novel operation strategy for wind farms' optimal Frequency Containment Reserve (FCR) provision that simultaneously distributes FCR and optimally controls wake formation. The power reserve allocation is dynamically decided at the wind farm supervisory control level, considering the intermittent wind power and direction, grid frequency stochasticity, and the aerodynamic complexity of the wake. A two-stage stochastic programming approach supports decision-making for an optimal contribution to day-ahead energy/FCR markets considering sub-hourly wind power and grid frequency uncertainty. Moreover, a novel method is used to reduce the computational complexity by employing a data-driven surrogate model of wake formation in the optimizer. This surrogate model consists of a neural network trained on the Gauss-Curl-Hybrid wake model in FLORIS. This deep learning approach allows fast estimation of the wake control parameters, i.e., the optimal yaw angles and axial induction factors. Then, a coevolutionary-based multi-objective particle swarm optimization searches for the optimal deloading of the WTs and maximizes the total power production and kinetic energy while minimizing wake. The performance of the proposed algorithm is evaluated on the C-Power wind farm layout in the North Sea. Simulation results demonstrate its effectiveness in improving the wind farm's overall performance for different operational conditions.

A two-stage stochastic programming approach for dynamic OD estimation using LBSN data

Estimating origin–destination (OD) demand is essential for urban transport management and traffic control systems. With the ubiquity of smartphones, location based social networks (LBSN) data has emerged as a new rich data source with broad urban spatial and temporal coverage highly suitable for OD estimation. Its nature of confirmed trip purpose (activity) and activity chain host makes it more advantageous than other data (e.g., household travel surveys and traffic network detection). On the other hand, LBSN data is a more direct and accurate representation of demand patterns and can remove the significant burden of developing traffic models and estimating simulation-based objective functions. However, thus far, most LBSN-based estimation models only focus on static (day-level) OD estimation, making less use of those characteristics. To this end, this paper establishes a two-stage stochastic programming (TSSP) framework integrating the activity chains to model activity-level dynamic mobility flows using LBSN data. The first-stage model aims to minimize the errors introduced by the inter-zone OD flows alongside the expected errors of the check-in patterns. The second-stage model attempts to minimize the errors produced by the considered check-in pattern scenarios. Markov chain Monte Carlo (MCMC) sampling is used to generate plausible check-in scenarios. A generalized Benders decomposition (GBD) algorithm is presented to solve the two-stage stochastic programming model. We conduct the experiments on the case study of Tokyo, Japan, under the employment of the generalized least squares (GLS) estimator. The results show that algorithm convergence can be guaranteed within several iterations. The approach can provide satisfactory estimations of check-in patterns, zonal production and attraction, and OD flows. Furthermore, multiple objective function states are tested for evaluating the completeness of the proposed framework and exploring its potential for simplification and extension. Incorporating specific penalty terms into the objective function also provides a way to verify the reliability of the two-stage structure and validate the effectiveness of the model. Finally, we discuss the model enhancement from the perspectives of online OD estimation by integrating with LBSN simulations, network-wide OD extrapolation using appropriate scaling methods, and removing the user-side data requirement by leveraging activity chain modeling. The proposed framework provides a novel and effective approach to OD demand estimation using LBSN data.

Optimization of crude oil operations scheduling by applying a two-stage stochastic programming approach with risk management

This paper focuses on the problem of crude oil operations scheduling carried out in a system composed of a refinery and a marine terminal, considering uncertainty in the arrival date of the ships that supply the crudes. To tackle this problem, we develop a two-stage stochastic mixed-integer nonlinear programming (MINLP) model based on continuous-time representation. Furthermore, we extend the proposed model to include risk management by considering the Conditional Value-at-Risk (CVaR) measure as the objective function, and we analyze the solutions obtained for different risk levels. Finally, to evaluate the solution obtained, we calculate the Expected Value of Perfect Information (EVPI) and the Value of the Stochastic Solution (VSS) to assess whether two-stage stochastic programming model offers any advantage over simpler deterministic approaches.

Carsharing equitable relocation problem: A two-stage stochastic programming approach with learning-embedded endogenous uncertainty in demand

Uncertain demand may exacerbate the imbalance of the supply–demand for a one-way carsharing system and complicate vehicle relocation decisions. To consider the effect of the uncertainty, this study proposes a two-stage stochastic nonlinear programming model, integrating long-term and short-term decisions and maximizing the profit of the carsharing companies. Specifically, in the first stage, tactical decisions of fleet sizing and initial vehicle distribution are determined before the realization of the uncertain demand. Operational decisions of both operator-based and user-based relocation are optimized in the second stage. Moreover, this paper first studies the user-based relocation incentives, which affect the distribution of uncertain demand with an endogenous relationship. A learning-embedded optimization method is introduced to learn such a distribution, enabling the decision-making optimization model to achieve higher performance under the guidance of the demand uncertainty. Second, we envision an equitable relocation issue that considers an uneven distribution of the unsatisfied demand with two different equity criteria measured from the aspects of stations and OD pairs, respectively. Third, the large problem scale, the nonlinear objective function and constraints, and the endogenous demand uncertainty constitute the nontrivial challenges to the feasible solution. For solving the problem efficiently, we linearize the nonlinear terms and develop a dedicated two-phase solution algorithm with a learning-embedded trust-region method in phase I to solve the continuous relaxation problem and a mixed-integer linear programming guided iterative rounding in phase II to obtain the integer solutions of carsharing operations. The solution algorithm adaptively bridges the learning and optimization process via the trust-region method with flexible sample generation. Finally, we conduct numerical experiments based on a real-world one-way carsharing system in Beijing to demonstrate the effectiveness and applicability of the proposed method and reveal some insights for the carsharing service.

A two-stage stochastic programming approach for planning of SVCs in PV microgrids under load and PV uncertainty considering PV inverters reactive power using Honey Badger algorithm

In order to reduce the Static Var Compensators’ (SVCs') initial investment costs and their anticipated yearly running costs, this paper offers a two-layer methodology for the allocation and sizing of the SVCs in radial distribution systems. First, the Power Loss (PLI) is used to choose the best candidate buses for allocating SVCs. A two-stage stochastic programming algorithm is then proposed to solve the sizing problem, with the Point Estimate Method (PEM) being used to account for the uncertainties in the PV and load powers. Analysis is also done on the advantages of combining the SVCs and the PV inverter's ability to provide reactive power. The optimization problem is solved using Honey Badger Algorithm (HBA) and compared to Archimedes Optimization Algorithm (AOA), Beetle antenna based Grey Wolf Optimization (BGWO), Particle Swarm Optimization (PSO) and Genetic Algorithm (GA). The IEEE 33 and 69 radial bus systems have been used to validate the suggested methodology. The findings demonstrate that, HBA models delivered the best outcomes in terms of convergence characteristics and computation time. When considering the PV inverter's capability to provide reactive power, estimated yearly operating costs are reduced by 17.44 % and 36.36 % for compensations of 600 kVAR and 1,000 kVAR, respectively.

Optimization of train schedule with uncertain maintenance plans in high-speed railways: A stochastic programming approach

In high-speed railways, unexpected disturbances on maintenance activities may cause serious delays of the scheduled trains and greatly affect the service quality for traveling passengers. In contrast to most existing studies that focused on deterministic maintenance activities, this paper develops a two-stage stochastic programming approach to address the optimization of train schedules under uncertain maintenance plans. Specifically, in the first stage, we aim to determine the departure times of trains from the origin station, since this information needs to be public to passengers in advanced. The objective function is to minimize the expected travel time of trains under uncertain duration time of maintenance activities. In the second stage, given the specific information of maintenance activities, we generate the train schedule by adjusting the stop patterns, train orders and the assignment of tracks at key stations. Due to the computational difficulties arising from the large number of discrete decision variables, we particularly develop a dual decomposition based solution approach to solve the two-stage stochastic model. Our approach decomposes the original problem into a set of scenario-dependent subproblems with much fewer number of variables, which greatly improves the computational efficiency. Finally, we conduct several sets of real-world instances based on the Beijing–Guangzhou high-speed railway corridor to verify the effectiveness of the proposed model and solution approach. The results demonstrate that our approach evidently outperforms state-of-art solvers (Gurobi), especially for large-scale instances that Gurobi cannot even return feasible solutions.

A Medium-Term Virtual Power Plant Optimization Framework Considering the Failure Rate of Its Intermittent Units Using Stochastic Programming

This article presents a model for maximizing the profit of a commercial virtual power plant (VPP) comprised of heterogeneous distributed energy resources (DERs) considering the failure or power outage of its intermittent units. The failures are taken into account through scenarios of failure happening in different parts of the units with different probabilities. The VPP has access to the future market, the day-ahead (DA) market, and bilateral contracts for trading. Since some of the parameters such as the DA market prices are volatile and uncertain, a two-stage stochastic programming approach is developed to simulate the uncertainty effectively. The VPP makes decisions regarding the future market and signing bilateral contracts in the first stage, then, decisions regarding trading in the DA market and the operation of the VPP’s DERs are taken in the second stage. Using the conditional value at risk (CVaR) approach, the behavior of the risk-neutral VPP is compared to the risk-averse VPP. It is shown that considering the failure of intermittent generation units of the VPP leads to more sensitivity of its profit toward pool prices for both risk-neutral and risk-averse VPPs. It also leads to at least 3.4% of VPP’s profit lost in the specific designed case study.

A Two-Phase Pattern Generation and Production Planning Procedure for the Stochastic Skiving Process

The stochastic skiving stock problem (SSP), a relatively new combinatorial optimization problem, is considered in this paper. The conventional SSP seeks to determine the optimum structure that skives small pieces of different sizes side by side to form as many large items (products) as possible that meet a desired width. This study studies a multiproduct case for the SSP under uncertain demand and waste rate, including products of different widths. This stochastic version of the SSP considers a random demand for each product and a random waste rate during production. A two-stage stochastic programming approach with a recourse action is implemented to study this stochastic N P -hard problem on a large scale. Furthermore, the problem is solved in two phases. In the first phase, the dragonfly algorithm constructs minimal patterns that serve as an input for the next phase. The second phase performs sample-average approximation, solving the stochastic production problem. Results indicate that the two-phase heuristic approach is highly efficient regarding computational run time and provides robust solutions with an optimality gap of 0.3% for the worst-case scenario. In addition, we also compare the performance of the dragonfly algorithm (DA) to the particle swarm optimization (PSO) for pattern generation. Benchmarks indicate that the DA produces more robust minimal pattern sets as the tightness of the problem increases.

Optimization under uncertainty of a hybrid waste tire and natural gas feedstock flexible polygeneration system using a decomposition algorithm

Market uncertainties motivate the development of flexible polygeneration systems that are able to adjust operating conditions to favor production of the most profitable product portfolio. However, this operational flexibility comes at the cost of higher capital expenditure. A scenario-based two-stage stochastic nonconvex Mixed-Integer Nonlinear Programming (MINLP) approach lends itself naturally to optimizing these trade-offs. This work studies the optimal design and operation under uncertainty of a hybrid feedstock flexible polygeneration system producing electricity, methanol, dimethyl ether, olefins or liquefied (synthetic) natural gas. A recently developed C++ based software framework (named GOSSIP) is used for modeling the optimization problem as well as its efficient solution using the Nonconvex Generalized Benders Decomposition (NGBD) algorithm. Two different cases are studied: The first uses estimates of the means and variances of the uncertain parameters from historical data, whereas the second assesses the impact of increased uncertain parameter volatility. The value of implementing flexible designs characterized by the value of the stochastic solution (VSS) is in the range of 260–405 M$ for a scale of approximately 893 MW of thermal input. Increased price volatility around the same mean results in higher expected net present value and VSS as operational flexibility allows for asymmetric exploitation of price peaks.

Two-stage stochastic programming approach for gas allocation network under uncertainty

Pipelines enable enormous amounts of various products (fluids) to be transported from supply nodes to demand nodes. They have traditionally been recognized as the most efficient and secure method of transferring gases. Uncertainties in the parameters may develop in the actual world for a variety of reasons which reduces the efficiency of the gas allocation network (GAN). The amount of available gas cannot be forecasted exactly due to the uncertainty in gas supply and shared usage by other demands. Design of GAN is carried out in a two stage manner: installation and operation. During operation, there is always a chance of change in number of sources or demands than that of initial design phase. In this paper, to deal with such discrete uncertainties, a two-stage stochastic programming approach for GAN is developed. The first stage represents the installation and commissioning of supplying nodes in order to satisfy the demands, and the second stage represents the actual allocation network under different available supply stations scenarios. Illustrative examples are presented to demonstrate the proposed solution procedure and annualized investment for the examples are calculated. The calculated annualized investment is 12%, 28% and 38% less than the worst case solutions. Result explains the benefits of the model in reducing investment costs while incorporating such discrete uncertainties.

A two-stage stochastic programming approach for reserving and allocating of emission trading permits under uncertainties

ABSTRACT The Emissions Trading Scheme (ETS) is a market-based approach aiming at reducing greenhouse gas (GHG) emissions. It involves the allocation of emission allowances to entities that are permitted to emit certain levels of GHG, as well as the re-sale and re-allocation of these allowances among entities. Considering the uncertainty of urban traffic congestion, the problem of reserving and allocating Emission Trading Permits (ETP) under stochastic demand is investigated. The ETP reserve of each link is determined before congestion, the optimal allocation scheme of ETP is determined after congestion. A two-stage stochastic programming model is formulated to minimize the sum of the reserve cost of ETP before congestion and the expected total loss of ETP after congestion. First, the effectiveness of the stochastic programming model is verified by comparing with the traditional method. Second, the total cost increases with the increase of initial reserve cost and shortage cost, and decreases with the increase of selling price and the capacity of links by sensitivity analysis. The unit reserve cost has a negative effect on the reserve quantity, while shortage cost, selling price, and link capacity have a positive effect on the reserve quantity. At last, the validity of the stochastic programming solution can be verified by numerical analysis of the Nguyen-Dupuis network. The stochastic programming solution proposed in this study shows an 89% reduction in ETP reserve compared to the optimal initial capacity solution. When level E occurs, the total cost is correspondingly reduced by 11%.

Optimal charging station locations and durations for a transit route with battery-electric buses: A two-stage stochastic programming approach with consideration of weather conditions

While the environmental advantages of battery-electric buses (BEBs) are well-known, their significant differences from diesel buses require alterations to both route design (i.e., charging station locations) and operations (i.e., schedule management and holding control). The location, number, charging duration, and types of charging stations must be considered as part of the planning process. Charging stations can be located at depots, termini, or en-route. This paper considers the long-term planning and optimization problem of en-route charging station locations and charging duration to optimize passengers’ waiting time and operation and capital costs while addressing the weather-induced stochasticity of ridership and battery performance of the BEBs. A linear deterministic optimization model and a two-stage stochastic programming (SP) optimization process are developed to place BEB charging stations along the route and estimate their assigned charging time for both one-way and two-way operations. The developed approaches are tested on two high-demand bus routes in Calgary. The impact of the breakdown of the charging station associated with the maximum charging time on the schedule and the cost of the BEB operation is assessed. The solution of the stochastic model is analyzed using the expected value of perfect information as an index. The results indicate that using the SP model helps decrease the expected travel time of the route while the total cost per trip increases compared to the deterministic model.

Stochastic regional-based profit-maximizing hub location problem: A sustainable overview

Climate change is one of the most important concerns globally, and countries need to improve their transportation infrastructure to handle flows with the least air pollution. In specific cities and regions, carbon emissions may cause in future many issues for people and societies. Although transportation hubs are very helpful in the reduction of transportation costs, they may cause an increase in carbon emissions in some regions, leading to long-term environmental problems. In this study, a model is developed for the profit-maximizing hub location problem that incorporates three pillars of sustainability. The proposed model is not only designed to maximize profit but also to control carbon emissions using a carbon cap policy and to reduce differences in emissions between regions with consideration of population density. This scheme will lead to achieving a sustainable transportation network. A two-stage stochastic programming approach is employed to cope with demand uncertainty. An enhanced sampling based on the self-organizing map method was utilized to cluster scenarios that lead to dealing with small-sized problems. Furthermore, classical Benders decomposition, Pareto-optimal cut Benders decomposition, and L-shaped algorithms are employed to solve the proposed models more efficiently. The proposed models are analyzed using the well-known Turkish Network (TR) data set. Computational results demonstrate that the proposed models help to achieve sustainability. Factors of sustainability such as environmental and social can be achieved by slightly reducing the amount of profit in the economic factor. Furthermore, results show that the L-shaped algorithm with a multi-cut scheme outperforms the commercial solver, classical Benders decomposition, and Pareto-optimal cut Benders decomposition algorithms in large-size instances.