Stage Stochastic Programming 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 uncertainty management in electricity markets with high renewable energy penetration

Abstract The high penetration of renewable energy sources (RESs) in modern power systems poses two conflicting issues. First one is the reduction in the operation costs resulted from RESs utilization instead of the expensive fossil fuel thermal generating units. However, the RESs are characterized by its uncertain and intermittent behavior that their output power is not dispatchable and not known exactly due to forecast errors. Therefore, reserves are scheduled in the day-ahead market to meet the uncertain supply from RESs which add some costs to the system. The decision maker should be aware of those two conflicting objectives in order to operate the system in the optimal way with minimum operation costs. The problem of the market clearing is formulated as Mixed Integer Linear Programming (MILP) problem using GAMS software. It is considered a two stage stochastic programming with the objective of minimizing the expected total operation energy and reserve costs while satisfying the various operational constraints. The results show the effectiveness of the RESs integration in different cases with the consideration of load shedding, RESs curtailment and transmission congestion. As shown from the results, when the network is congested, the operation costs are increased due to the load shedding and RESs curtailment events occur. Moreover, scheduled reserves are increased to face the uncertainty of RESs. As the RESs power penetration increases with no network congestion, a decreasing trend in the operation cost can be seen. However, this trend is less marked as the uncertainty of RESs generation increases. For the network congested case, the reduction rate of the operation cost is decreased for increasing RESs power penetration levels.

Two-Stage Stochastic Optimization Model for Multi-Microgrid Planning

This paper presents a Two Stage stochastic Programming (TSSP) model for the planning of Multi-Microgrids (MMGs) in Active Distribution Networks (ADNs). The model aims to minimize the total costs while benefiting from interconnections of Microgrids (MGs), considering uncertainties associated with electricity demand and Renewable Energy Sources (RESs). The associated uncertainties are analyzed using Geometric Brownian Motion (GBM) and probability distribution functions (pdfs). The model includes long-term purchase decisions and short-term operational constraints, using Geographical information Systems (GIS) to realistically estimate rooftop solar limits. The planning model is used to study the feasibility of implementing an MMG system consisting of 4 individual Microgrids (MGs) at an ADN in a municipality in the state of São Paulo, Brazil. The results show that the TSSP model tends to be less conservative than the deterministic planning model, which is based on simple and pessimistic reserve constraints, while performing faster than a simple Stochastic Linear Programming (SLP) algorithm, with higher accuracy.

Stochastic Coded Offloading Scheme for Unmanned-Aerial-Vehicle-Assisted Edge Computing

Unmanned aerial vehicles (UAVs) have gained wide research interests due to their technological advancement and high mobility. The UAVs are equipped with increasingly advanced capabilities to run computationally intensive applications enabled by machine learning techniques. However, because of both energy and computation constraints, the UAVs face issues hovering in the sky while performing computation due to weather uncertainty. To overcome the computation constraints, the UAVs can partially or fully offload their computation tasks to the edge servers. In ordinary computation offloading operations, the UAVs can retrieve the result from the returned output. Nevertheless, if the UAVs are unable to retrieve the entire result from the edge servers, i.e., straggling edge servers, this operation will fail. In this article, we propose a coded distributed computing (CDC) approach for computation offloading to mitigate straggling edge servers. The UAVs can retrieve the returned result when the number of returned copies is greater than or equal to the recovery threshold. There is a shortfall if the returned copies are less than the recovery threshold. To minimize the cost of the network, energy consumption by the UAVs, and prevent over and under subscription of the resources, we devise a two-phase stochastic coded offloading scheme (SCOS). In the first phase, the appropriate UAVs are allocated to the charging stations amid weather uncertainty. In the second phase, we use the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$z$ </tex-math></inline-formula> -stage stochastic integer programming (SIP) to optimize the number of computation subtasks offloaded and computed locally, while taking into account the computation shortfall and demand uncertainty. By using a real data set, the simulation results show that our proposed scheme is fully dynamic and minimizes the cost of the network and UAV energy consumption amid stochastic uncertainties.

Inventory availability commitment under uncertainty in a dropshipping supply chain

Dropshipping, where e-retailers manage marketing and dropshippers manage inventory and fulfillment, has become a common practice in e-commerce. However, due to information asymmetry, some e-retailers require dropshippers to share available inventory quantity committed to them on a periodic basis (x-hour, day,...), via an availability commitment. Such commitment leaves the potential for dropshippers to overpromise and exploit availability pooling benefit as it does not involve physical transaction. To further prevent fulfillment failure, some e-retailers stop accepting more orders once remaining promised quantity falls below a promised availability threshold, which is unknown to the dropshipper. In this paper, leveraging the collaboration with a dropshipping furniture manufacturer in the US market, we tackle the inventory availability commitment (IAC) problem for dropshippers at the operational decision level. Three commitment policies are proposed based on varying overpromising allowance: guaranteed fulfillment, controlled fillrate, and penalty-driven fillrate policies. The IAC optimization problem, under uncertain customer demand and retailers threshold, is modeled as a two stage stochastic program. Experimental results on a case study demonstrated that penalty-driven fillrate policy is a dominating policy for a dropshipper under any availability level while the impact of overpromising is maximized under lean availability. E-retailers imposing thresholds tend to receive more availability than those not imposing any threshold in general. However, non-overpromising dropshippers with lean or lower availability level will promise less availability to e-retailers imposing threshold.

Value of Operational Flexibility in Co-Production Systems with Yield and Demand Uncertainty

We study the production decisions of a firm that operates in a co-production system (single input but multiple simultaneous outputs) with random yield and demand. The firm uses its outputs to meet multiple end-market demands with different quality requirements. Outputs serving a market segment may be a result of a blending process of different quality products of the co-production system. After the realization of yield and before the realization of demand, the firm has the option to upgrade the quality of outputs in order to better position itself to meet market demands. With the goal of maximizing expected profits, we formulate a three stage stochastic programming problem, to investigate the value of intermediate upgrading flexibility. Using a stylized model of two products and two markets with different quality requirements, we characterize the optimal decisions and show that the quality upgrading policy is of a single-threshold type. Although upgrading is costly, it creates value for the firm through reducing total production cost and better managing the yield uncertainty. Positive correlation between demands of the end-markets decreases the value of upgrading. The more general model formulation offers the blueprint of a stochastic programming model that can be solved for realistic applications.

A Methodology for Explicit Representation Of The Stochastic Demand Due To Electric Vehicles in Generation Expansion Planning Problems

Generation expansion planning (GEP) problems are solved to find the optimum investment decisions to satisfy the increasing electricity demand. Integration of electric vehicles (EVs) with the capability of charging from the grid will also increase the electricity demand of the grid. Depending on the charging/driving characteristics of users, demand curves for EVs will be shaped and it will be different on each day. Therefore, it is very crucial to represent this stochastic nature of EVs demand in the associated GEP problems. This paper is proposing a methodology to represent EVs demand realistically on GEP models. The proposed methodology starts with generating random demand patterns to demonstrate possibilities for the EVs demand patterns via Monte Carlo Simulation, then using an optimization-based model to select a representative set. Two stage stochastic programming model is proposed for GEP problems and solved to minimize the expected cost over the entire set, the representative set and the average EVs demand. The results show that GEP models with selected demand curves produce more realistic decisions (closer to the solutions obtained by using the entire demand patterns) than the decisions obtained by the models with average EVs demand. In most cases, the models using average EVs demand fail to capture the new peaks generated by EVs, therefore, they suggest less capacity expansion then the required amount. This results in more unmet demand in the system.

A two stage stochastic programming for asset protection routing and a solution algorithm based on the Progressive Hedging algorithm

In this paper, a two-stage stochastic programming model is developed for the asset protection routing problem (APRP) to be employed in anticipation of an escaped wildfire. In this model, strategic and tactical decisions are considered in a two-stage setting. The locations of protection depots are determined, taking into account the routing decisions under different possible scenarios. To solve the proposed model, the Frank–Wolfe Progressive Hedging decomposition approach is employed. A realistic case study set in south Hobart, Tasmania, is considered. In this study, the scenarios for uncertain parameters are generated based on real data, considering different sources of uncertainties such as wind direction and speed and total monthly rainfall. Computational experiments have been conducted to demonstrate the solution algorithm’s efficiency in solving the asset protection routing problem with a two-stage stochastic framework. The numerical results suggest that more assets with higher values can be protected by considering the proposed two-stage stochastic programming model. The value of the approach is particularly significant where resources are limited, and uncertainty levels are high. Moreover, the model and solution procedure can be applied to other disaster situations in which protection activities occur.

Using revenue management technique to allocate the capacity in reliable hub network design under uncertain air passenger traffic

This paper introduces a two stage stochastic programming to address strategic hub location decisions and tactical flight routes decisions for various customer classes considering uncertainty in demands. We considered the airline network with the arc capacitated single hub location problem based on complete–star p-hub network. In fact, the flight routes are allowed to stop at most two different hubs. The first stage of the model (strategic level) determines the network configuration, which does not change in a short space of time. The second stage is dedicated to specify a service network consists of determining the flight routes and providing booking limits for all itineraries and fare classes after realization of uncertain scenarios. To deal with the demands uncertainty, a stochastic variations caused by seasonally passengers’ demands through a number of scenarios is considered. Since airline transportation networks may face different disruptions in both airport hubs and communication links (for example due to the severe weather), proposed model controls the minimum reliability for the network structure. Due to the computational complexity of the resulted model, a hybrid algorithm improved by a caching technique based on genetic operators is provided to find a near optimal solution for the problem. Numerical experiments are carried out on the Turkish network data set. The performance of the solutions obtained by the proposed algorithm is compared with the pure GA and Particle Swarm Optimization (PSO) in terms of the computational time requirements and solution quality.

Impact of Long-Term Water Inflow Uncertainty on Wholesale Electricity Prices in Markets with High Shares of Renewable Energies and Storages

Renewable energy shares in electricity markets are increasing and therefore also require an increase in flexibility options. Conventional electricity price modelling with optimisation models in thermally dominated markets is not appropriate in markets with high shares of renewable energies and storages because price structures are not adequately represented. Previous research has already identified the impact of uncertainty in renewable energy feed-in on investment and dispatch decisions. However, we are not aware of any work that investigates the influence of uncertainties on price structures by means of optimisation models. Appropriate modelling of electricity price structures is important for investment and policy decisions. We have investigated the influence of uncertainty concerning water inflow by applying a second stage stochastic dual dynamic programming approach in a linear optimisation model using Norway as an example. We found that the influence of uncertainty concerning water inflow combined with high shares of storages has a strong impact on the electricity price structures. The identified structures are highly influenced by seasonal water inflow, electricity demand, wind, and export profiles. Additionally, they are reinforced by seasonal primary energy source prices and import prices. Incorporating uncertainties in linear optimisation models improves the price modelling and provides, to a large extent, an explanation for the seasonal patterns of Norwegian electricity market prices. The paper explains the basic pricing mechanisms in markets with high shares of storages and renewable energies which are subject to uncertainty. To identify these fundamental mechanisms, we focused on uncertainty regarding water inflow, but the basic results hold true for uncertainties regarding other renewable energies as well.

A risk-based fuzzy boundary interval two-stage stochastic water resources management programming approach under uncertainty

Establishing an optimal water allocation scheme is a challenging yet crucial task for effective risk management of water resources. The dynamic and complex interaction among various risk-related components in a water resources management system cannot be easily modeled to characterize the system response under different risk control settings. In this study, a risk-based fuzzy boundary interval two-stage stochastic water resources management programming (RFITSWMP) model is developed for the optimization of water consumption. This model incorporates a series of risk control constraints, such as water availability, maximum allowable penalty, and allowable benefit violation constraints into a fuzzy boundary interval two-stage stochastic programming framework for water resources management. It can address the uncertainties presented as fuzzy boundary intervals and probability distributions. It can also tackle the recourse action to minimize penalties based on interactive influences of different risk control measures, further generating optimal water allocation alternatives and guiding water resources management. This developed model is applied to a case study of water consumption optimization in the middle reaches of the Heihe River Basin in China. Feasible water allocation schemes under given risk levels, as well as the associated economic benefit, actual benefit and penalty loss are generated. The results can help decision makers to gain an insight into the inherent conflicts and tradeoffs amid risk, benefit and water allocation. The performance of developed model is further demonstrated by comparing it with a fuzzy boundary interval two stage stochastic water resources management programming (FITSWMP) model. In addition, a multiple factorial analysis (MFA) approach is employed to analyze the impacts of interactive risk parameters on the optimal decisions. The result disclosed that (1) the higher individual risk-parameter levels correspond to higher water shortage, penalty loss and benefit value-at-risk but higher economic benefit. (2) The combination of risk parameters can achieve robust water allocation and economic benefit as the constraints concerned with risk parameters affect and limited by each other. (3) The condition that water violated probability (p) is 0.2, penalty violated risk level (β) is 1 and benefit violated risk level (γ) is 1 has the highest water allocation and biggest economic benefit.

A generalized Benders decomposition-based branch and cut algorithm for two-stage stochastic programs with nonconvex constraints and mixed-binary first and second stage variables

In this paper, we propose a generalized Benders decomposition-based branch and cut algorithm for solving two stage stochastic mixed-integer nonlinear programs (SMINLPs) with mixed binary first and second stage variables. At a high level, the proposed decomposition algorithm performs spatial branch and bound search on the first stage variables. Each node in the branch and bound search is solved with a Benders-like decomposition algorithm where both Lagrangean cuts and Benders cuts are included in the Benders master problem. The Lagrangean cuts are derived from Lagrangean decomposition. The Benders cuts are derived from the Benders subproblems, which are convexified by cutting planes, such as rank-one lift-and-project cuts. We prove that the proposed algorithm converges in the limit. We apply the proposed algorithm to a stochastic pooling problem, a crude selection problem, and a storage design problem. The performance of the proposed algorithm is compared with a Lagrangean decomposition-based branch and bound algorithm and solving the corresponding deterministic equivalent with the solvers including BARON, ANTIGONE, and SCIP.

The urban recharging infrastructure design problem with stochastic demands and capacitated charging stations

In this study we develop an exact solution method to optimize the location and capacity of charging stations to satisfy the fast charging needs of electric vehicles in urban areas. Stochastic recharge demands, capacity limitations of charging stations and drivers’ route preferences (deviation tolerances) are simultaneously considered to address this challenging problem faced by recharging infrastructure planners or investors. Taking a scenario based approach to model demand uncertainty, we first propose a compact two stage stochastic programming formulation. We then project out the second stage decision variables from the compact formulation by describing the extreme rays of its polyhedral cone and obtain (1) a cut formulation that enables an efficient branch and cut algorithm to solve large problem instances (2) a novel characterization for feasible solutions to the capacitated covering problems. We test our algorithm on the Chicago metropolitan area network, by considering real world origin-destination trip data to model charging demands. Our results attest the efficiency of the proposed branch and cut algorithm and provide significant managerial insights.

Quantitative stability analysis for minimax distributionally robust risk optimization

This paper considers distributionally robust formulations of a two stage stochastic programming problem with the objective of minimizing a distortion risk of the minimal cost incurred at the second stage. We carry out a stability analysis by looking into variations of the ambiguity set under the Wasserstein metric, decision spaces at both stages and the support set of the random variables. In the case when the risk measure is risk neutral, the stability result is presented with the variation of the ambiguity set being measured by generic metrics of $$\zeta $$ -structure, which provides a unified framework for quantitative stability analysis under various metrics including total variation metric and Kantorovich metric. When the ambiguity set is structured by a $$\zeta $$ -ball, we find that the Hausdorff distance between two $$\zeta $$ -balls is bounded by the distance of their centers and difference of their radii. The findings allow us to strengthen some recent convergence results on distributionally robust optimization where the center of the Wasserstein ball is constructed by the empirical probability distribution.

New decision support system for strategic planning in process industries: Computational results

Abstract The impact of a Stochastic Linear Programming (SLP) based Decision Support System in a manufacturing company, such as an integrated aluminum plant, is measured by two important parameters, the VSS and EVPI. With the real data of an integrated steel plant in India, we demonstrate that SLP based DSS can be very effective in managing demand uncertainty and performing futuristic integrated planning, and their financial impact can be in millions of dollars. A two stage stochastic programming model with recourse is implemented in the DSS here. A set of experiments is conducted. Real data from an aluminum company is used to validate the system. The importance of SLP based DSS can be realized from the fact that the value of the stochastic solution (VSS) is USD 3.58 million with 30% demand variability and equally likely demand distribution. The VSS as a percentage of Expectation of Expected Value (EEV) ranges from 0.90% to 18.93% across experiments.

Moment-SOS relaxation of the medium term hydrothermal dispatch problem

This paper analyzes the application of moment-SOS (sum of squares) relaxation to solve the medium term hydrothermal dispatch (MTHD) problem for predominantly hydro systems. The optimal dispatch is formulated as a two stage stochastic programming problem, which is nonconvex due to constraints that represent hydro power production, and may have multiple solutions. The size and conditioning of the problem pose challenges to the use of the semidefinite programming (SDP) techniques that are discussed in the text. The size of the problem restricts this study to first order moment relaxations. Therefore, to improve the quality of relaxed solutions, additional meaningful constraints are introduced in the SDP problems. Subsequently, the problem re-scalded to achieve numerical stability in SDP relaxation. Results obtained for different test systems, most of them being partitions of the Brazilian hydroelectric system, indicate that the procedures used to improve the quality of SDP solutions are effective.

Transportation and Production Lot-size for Sugarcane under Uncertainty of Machine Capacity

The integrated transportation and production lot size problems is important effect to total cost of operation system for sugar factories. In this research, we formulate a mathematic model that combines these two problems as two stage stochastic programming model. In the first stage, we determine the lot size of transportation problem and allocate a fixed number of vehicles to transport sugarcane to the mill factory. Moreover, we consider an uncertainty of machine (mill) capacities. After machine (mill) capacities realized, in the second stage we determine the production lot size and make decision to hold units of sugarcane in front of mills based on discrete random variables of machine (mill) capacities. We investigate the model using a small size problem. The results show that the optimal solutions try to choose closest fields and lower holding cost per unit (at fields) to transport sugarcane to mill factory. We show the results of comparison of our model and the worst case model (full capacity). The results show that our model provides better efficiency than the results of the worst case model.

Value of operational flexibility in co-production systems with yield and demand uncertainty

We study the production decisions of a firm that operates a co-production system (single input but multiple simultaneous outputs) with random yield and demand. The firm uses its outputs to meet multiple end-market demands with different quality requirements. Outputs serving a market segment may be a result of a blending process of different quality products of the co-production system. After the realisation of yield and before the realisation of demand, the firm has the option to upgrade the quality of outputs in order to better position itself to meet market demands. With the goal of maximising expected profits, we formulate a three stage stochastic programming problem, to investigate the value of intermediate upgrading flexibility. Using a stylised model of two products and two markets with different quality requirements, we characterise the optimal decisions and show that the quality upgrading policy is of a single-threshold type. Although upgrading is costly, it creates value for the firm through reducing total production cost and better managing the yield uncertainty. Positive correlation between demands of the end-markets decreases the value of upgrading. The more general model formulation offers the blueprint of a stochastic programming model that can be solved for realistic applications.

A stochastic program approach for path reconstruction oriented sensor location model

Path flow identification is of particular interest for a number of traffic applications, such as OD demand estimation, link flow inference, and toll freeway revenue management. Optimal positioning of active sensors can help to identify path flows. Due to the stochastic nature of transportation systems, we propose a scenario based two stage stochastic programming framework which considers the uncertainty of the link-path matrix. The first stage model aims to minimize the total traffic sensor installation cost and the expected penalty for uncovered and undifferentiated paths. The second stage model attempts to minimize uncovered and undifferentiated paths for a given sensor location pattern and a specific scenario. In addition, a mean risk measure is also incorporated into the two stage stochastic programming framework, and consequently a mean risk two stage stochastic programming model is proposed. Both models have the same structure, where the first stage and second stage decision variables are binary. The second stage decision variable can be relaxed to a continuous variable without changing the nature of the model. To solve the two stochastic programming models, a branch and bound based integer L-shaped algorithm is presented. Finite steps convergence is guaranteed for the algorithm. To handle the problem with a large number of scenarios, a sampling technique is introduced, and the confidence bound is analyzed with respect to the scenario size. Extensive numerical experiments are conducted to verify the effectiveness of the proposed models and algorithm. The most important numerical results are as follows: (i) the stochastic programming framework is capable of capturing the reality more efficiently and accurately, (ii) the path differentiation factor is more critical than the path coverage factor in determining the sensor placement pattern, and (iii) in the partial parameter setting case, the mean risk based stochastic programming model results in a significantly different sensor placement pattern compared to the normal stochastic programming model. The study contributes to practical sensor placement design.

A two-stage stochastic programming model for rail-truck intermodal network design with uncertain customer demand

With the globalization of supply chains, intermodal operations have become a key player in transportation. In real life, the consumer’s demands are often uncertain. The recent literature is rich of papers that either emphasizes on uncertain customer demands or intermodal transportation planning. In this paper, we combine these two fields in order to identify new potentials for saving cost and improving the efficiency of intermodal logistics network. A two stage stochastic programming model was proposed to decide the terminal location and transportation routes under uncertain customer demands. And the AP data set is used to test the presented model. The result is compared with the traditional counterparts, which shows superiority in solution quantity under uncertain environment.