Stochastic Mathematical Programming Research Articles

Adaptive Simultaneous Stochastic Optimization of the Escondida Mining Complex, Chile

AbstractThis paper presents the application of adaptive simultaneous stochastic optimization with a representative branching framework to generate a strategic mining plan for the Escondida mining complex, the world’s largest copper-production operation. This adaptive stochastic optimization considers geological uncertainty while integrating investment and operational alternatives in the production schedule of a mining complex. Mining complexes are comprised of interconnected components affected by multiple sources of uncertainty. Thus, they must be optimized simultaneously in order to maximize their value, manage environmental impacts, and minimize risk. Additionally, due to the extensive lives of assets and the dynamic and uncertain environment in which mining complexes operate, it is not reasonable to assume that the current strategic plan will remain optimal. Thus, an operationally feasible method to embed alternatives in the mine plan is used. The method utilized provides a strategic plan with representative branches for future possible investment decisions. Adaptive decisions are made sequentially over time, activating costs and effects over the model. The optimization process chooses the optimal strategic production plan accordingly, as well as the investments made and their timing. The Escondida mining complex is a multi-element, multi-pit operation with nine different processing destinations. Investment options considered herein are truck and shovel fleet sizing, adding a secondary crusher in one of the plants, and investing in a main crusher assigned to one of the pits. Additionally, operational alternatives at the mine and plant levels are included. The adaptive solution shows a substantial probability that the mine plan might change its design substantially due to geological uncertainty, presenting an increased expected net present value when compared to the previously developed stochastic mathematical programming formulation that does not consider adaptive decisions, thus generating a single static strategic production plan for the related mining complex. Further studies at the Escondida mining complex can consider adoptive solutions integrating capital investments pertinent to climate change issues.

Simultaneous stochastic optimisation of mining complexes: Integrating progressive reclamation and waste management with contextual bandits

ABSTRACT Managing environmental performance of waste dump facilities in mining complexes is an integral part of long-term production planning. Sustainable long-term production scheduling solutions are desired to mitigate risk and return the environment to a productive post-mining state. A simultaneous stochastic optimisation framework for long-term production scheduling in mining complexes is developed that integrates waste management and progressive reclamation. The waste dump placement schedule is jointly optimised with the extraction sequence, destination policy, and stockpiling decisions in a single stochastic mathematical programming framework. This includes the timing of progressive reclamation activities in parallel with production to enhance waste dump rehabilitation. Uncertainty related to the production of acid rock drainage is quantified by simulating geochemical properties of waste and managing the blending of uncertain waste properties within the optimisation framework. With respect to the framework for simultaneous stochastic optimisation, contextual bandits are explored to improve the metaheuristic solution approach and solve the corresponding large-scale optimisation model. The framework is tested in a multi-mine copper-gold mining complex leading to improved environmental performance. Risk of acid rock drainage is decreased by 52.5% in the waste dump facilities. Reclamation planning activities for meeting environmental requirements are scheduled prior to closure. The solution approach more effectively improves the objective function with contextual bandits leading to a 24% improvement in the study presented.

Entropy-metric estimation of the small data models with stochastic parameters

The formalization of dependencies between datasets, taking into account specific hypotheses about data properties, is a constantly relevant task, which is especially acute when it comes to small data. The aim of the study is to formalize the procedure for calculating optimal estimates of probability density functions of parameters of linear and nonlinear dynamic and static small data models, created taking into account specific hypotheses regarding the properties of the studied object. The research methodology includes probability theory and mathematical statistics, information theory, evaluation theory, and stochastic mathematical programming methods. The mathematical apparatus presented in the article is based on the principle of maximization of information entropy on sets determined as a result of a small number of censored measurements of “input" and “output" entities in the presence of noise. These data structures became the basis for the formalization of linear and nonlinear dynamic and static models of small data with stochastic parameters, which include both controlled and noise-oriented input and output measurement entities. For all variants of the above-mentioned small data models, the tasks of determining the optimal estimates of the probability density functions of the parameters were carried out. Formulated optimization problems are reduced to the forms canonical for the stochastic linear programming problem with probabilistic constraints.

Connecting planning horizons in mining complexes with reinforcement learning and stochastic programming

Connecting short- and long-term production schedules in mining complexes is essential to ensure that the long-term production schedule is achievable at shorter timescales. Previous research that addresses optimizing mining complexes under uncertainty focus on simultaneously optimizing different components in the mining complex to capitalize on advantageous synergies. Typically, short- and long-term production schedules are optimized separately in a number of stages. This poses risk of schedule misalignment, which can adversely affect the economic outcome of a mining complex and the ability to meet long-term production forecasts at shorter timescales. A framework is proposed to jointly optimize short- and long-term production schedules by connecting planning horizons with stochastic mathematical programming and reinforcement learning. The solution approach is tested in a large operating copper mining complex and demonstrates significant improvements in the resulting production and financial forecasts.

Research on Railway Emergency Resources Scheduling Model under Multiple Uncertainties

This paper discusses the optimization of emergency resource scheduling for major railway emergencies under multiple uncertainties while considering the uncertainties in demand, reserve, and transportation costs of resources. We introduce a novel approach that integrates stochastic mathematical programming, interval parameter programming, and fuzzy mathematical programming to study uncertain parameter interactions and coupling. A two-stage interval fuzzy credibility-constrained model is established and solved using an interval interactive algorithm. Finally, through a case study on China Railway Nanchang Group Co., Ltd., the novelty and effectiveness of the proposed method for optimizing emergency resource scheduling strategies under multiple uncertainties are demonstrated.

Coordinated expansion planning of transmission and distribution systems integrated with smart grid technologies

Integration of smart grid technologies in distribution systems, particularly behind-the-meter initiatives, has a direct impact on transmission network planning. This paper develops a coordinated expansion planning of transmission and active distribution systems via a stochastic multistage mathematical programming model. In the transmission level, in addition to lines, sitting and sizing of utility-scale battery energy storage systems and wind power plants under renewable portfolio standard policy are planned. Switchable feeders and distributed generations are decision variables in the distribution level while the impact of demand response programs as a sort of behind-the-meter technologies is accommodated as well. Expansion of electric vehicle taxi charging stations is included as a feasible option in both transmission and distribution levels. In order to deal with short-term uncertainty of load demand, renewable energy sources output power, and the charging pattern of electric vehicle taxis in each station, a chronological time-period clustering algorithm along with Monte Carlo simulation is utilized. The proposed model is tackled by means of Benders Dual Decomposition (BDD) method. The IEEE RTS test system (as the transmission system) along with four IEEE 33-node test feeders (as distribution test systems) are examined to validate effectiveness of the proposed model.

On the computation of equilibria in monotone and potential stochastic hierarchical games

We consider a class of noncooperative hierarchical $${\textbf{N}}$$ -player games where the ith player solves a parametrized stochastic mathematical program with equilibrium constraints (MPEC) with the caveat that the implicit form of the ith player’s in MPEC is convex in player strategy, given rival decisions. Few, if any, general purpose schemes exist for computing equilibria, motivating the development of computational schemes in two regimes: (a) Monotone regimes. When player-specific implicit problems are convex, then the necessary and sufficient equilibrium conditions are given by a stochastic inclusion. Under a monotonicity assumption on the operator, we develop a variance-reduced stochastic proximal-point scheme that achieves deterministic rates of convergence in terms of solving proximal-point problems in monotone/strongly monotone regimes with optimal or near-optimal sample-complexity guarantees. Finally, the generated sequences are shown to converge to an equilibrium in an almost-sure sense in both monotone and strongly monotone regimes; (b) Potentiality. When the implicit form of the game admits a potential function, we develop an asynchronous relaxed inexact smoothed proximal best-response framework, requiring the efficient computation of an approximate solution of an MPEC with a strongly convex implicit objective. To this end, we consider an $$\eta $$ -smoothed counterpart of this game where each player’s problem is smoothed via randomized smoothing. In fact, a Nash equilibrium of the smoothed counterpart is an $$\eta $$ -approximate Nash equilibrium of the original game. Our proposed scheme produces a sequence and a relaxed variant that converges almost surely to an $$\eta $$ -approximate Nash equilibrium. This scheme is reliant on resolving the proximal problem, a stochastic MPEC whose implicit form has a strongly convex objective, with increasing accuracy in finite-time. The smoothing framework allows for developing a variance-reduced zeroth-order scheme for such problems that admits a fast rate of convergence. Numerical studies on a class of multi-leader multi-follower games suggest that variance-reduced proximal schemes provide significantly better accuracy with far lower run-times. The relaxed best-response scheme scales well with problem size and generally displays more stability than its unrelaxed counterpart.

Designing Watersheds for Integrated Development (DWID): A stochastic dynamic optimization approach for understanding expected land use changes to meet potential water quality regulations

This study investigates the tradeoff between agricultural economic losses and potential water quality regulations at the watershed level over time. Specifically, we applied a stochastic dynamic mathematical programming approach to evaluate the impact of land use changes on agricultural economic losses under uncertain water quality regulations and crop yields. We selected the Little River Experimental Watershed in South Georgia as a case study. Seven potential water pollution reduction trajectories for total phosphorus (TP) were developed. The proposed model applied a chance-constraint technique to allow some degree of uncertainty in meeting the water quality standards. The result shows that agricultural profits decrease when uncertainties are considered at the watershed level. The findings further show that uncertainties of meeting the water quality standard expedited the land use transition from croplands to forestlands and magnified the share of forestlands, highlighting the crucial role of forests in managing water pollution in the selected watershed. The results also indicate that under a moderate pollution reduction scenario, an economic loss of $232/ha/yr is expected to achieve a 16 % reduction in TP concentration. In a more ambitious pollution mitigation scenario, agricultural economic losses are expected to be $778/ha/yr and may result in a 45 % decrease in TP concentration. The modeling approach developed in Designing Watersheds for Integrated Development (DWID) could easily feed into those policy deliberations which operate at the interface of economics and water-related policies for ensuring sustainable management of watersheds worldwide.

Green port oriented resilience improvement for traffic-power coupled networks

Under the green ports initiative, port operations rely more on electric power. However, the power system is vulnerable to disruptions. Its ability to resist disruption seriously affects port resilience, which is related to the port operation’s level and performance. To maintain a resilient port, its infrastructure needs to be well designed and coordinated to prevent or reduce losses from disruptions. Therefore, some policies are proposed to improve port resilience under various disruption scenarios by focusing on the traffic-electric power coupled system and optimizing the planning level and operational level decisions. A two-stage, stochastic, and nonlinear mathematical programming model is established to model systems’ interaction and describe the impacts of preparedness and recovery actions with the objective of maximizing port resilience under a certain budget. The experimental results show that the improved coupled system can greatly improve the port resilience. In addition, the resilience of the port can also be improved by optimizing the investment in the power network and the equipment scheduling in the container transportation network. Some forward-looking managerial implications have been revealed.

A Mathematical programming approach for a vessel scheduling-transportation problem with multiple sources and destinations, and normal daily demand distributions

The principal focus of this research effort is to investigate a maritime stochastic transportation problem. In this case, crude oil shipments are to be transported from multiple sources to multiple destinations. The demand at the destinations is normally distributed and the violation of certain specified lower and upper daily storage limits can lead to various types of penalties. We aim to utilize (stochastic) mathematical programming approaches to minimize the overall cost of such fleet operation by optimizing the vessel schedules and maintaining acceptable daily storage levels considering the stochastic demand structures. This research effort signifies that the nature of daily demand distributions in the multiple sources-destinations scheduling-inventory scenario significantly impacts the overall fleet schedules and the total expected cost. Therefore, it is crucial to grasp essential stochastic aspects of the daily demands to avoid potential misrepresentation of the operational costs. The robustness of the adopted approach is illustrated by presenting computational results that are based on a wide range of test problems. Moreover, our computational study also examines the impact of variations in demand and the probability of meeting demands on the cost structures.

Strategic Offering of a Price Maker Wind Power Producer in Distribution-Level Energy Markets in Presence of Flexible Prosumers

This paper presents an optimal bidding strategy for a strategic wind power producer (WPP) in a distribution-level energy market (DLEM). The behavior of the WPP is modelled through a bi-level stochastic optimization problem where the upper-level problem maximizes the profit of the WPP and the lower-level problem describes the clearing processes of the DLEM while considering network constraints. The bi-level problem is a stochastic mathematical program with equilibrium constraints (MPEC) that is formulated as a mixed-integer linear programming (MILP) problem. The main focus of this study is investigating prosumers’ impact on the market power of the strategic WPP in a DLEM structure. In this model, the effect of flexible prosumers from the aspects of demand response (DR) participants and photovoltaic penetration level (PVPL) on the WPP’s offering strategy is investigated. Moreover, the impact of bilateral contract on the market power of the strategic WPP and the cleared prices of the network is addressed. The proposed model is implemented in an IEEE 33-bus and numerical results illustrate how behavior of flexible prosumers and PVPL index affect the decision making of the strategic WPP when network constraints are considered. Numerical results show that by active participation of prosumers in DR programs, the reliance of DLEM on the strategic WPP reduces. Moreover, if the WPP participates in bilateral contracts, its offering to the DELM decreases, and as the result, the cleared prices augment indicating market power of the WPP.

Smart control of air conditioning systems in manufacturing systems facing uncertainty

Enhancing energy use efficiency and reducing energy demand are important issues for industry and academe. Studies on the ventilation, and air conditioning (HVAC) system highlighted the optimization of individual devices, but did not fully consider global optimization. The heating and cooling mechanism of an HVAC system in the present study comprises multiple chillers, chilled water pumps, condenser water pumps, and cooling towers. This study proposes a smart control model of heating, ventilation, and HVAC systems in manufacturing systems to deal with dynamic indoor cooling demands and outdoor temperature changes. A stochastic mixed integer mathematical programming model optimizes the heating and cooling decisions of the HVAC system for energy conservation. This study also investigates a case of the HVAC system in a TFT-LCD manufacturing firm. Experiment results indicate that the proposed model saves up to 3.2% of total energy in the target HVAC system.

Optimal Price-maker Trading Strategy of Wind Power Producer Using Virtual Bidding

This paper proposes a stochastic optimization model for generating the optimal price-maker trading strategy for a wind power producer using virtual bidding, which is a kind of financial tool available in most electricity markets of the United States. In the proposed model, virtual bidding is used to improve the wind power producer's market power in the day-ahead (DA) market by trading at multiple buses, which are not limited to the locations of the wind units. The optimal joint wind power and virtual trading strategy is generated by solving a bi-level nonlinear stochastic optimization model. The upper-level problem maximizes the total expected profit of the wind power and virtual bidding while using the conditional value at risk (CVaR) for risk management. The lower-level problem represents the clearing process of the DA market. By using the Ka-rush-Kuhn-Tucker (KKT) conditions, duality theory, and big- <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$M$</tex> method, the bi-level nonlinear stochastic model is firstly transferred into an equivalent single-level stochastic mathematical program with the equilibrium constraints (MPEC) model and then a mixed-integer linear programming (MILP) model, which can be solved by existing commercial solvers. To reduce the computational cost of solving the proposed stochastic optimization model for large systems, a method of reducing the number of buses considered for virtual bidding is proposed to simplify the stochastic MPEC model by reducing its decision variables and constraints related to virtual bidding. Case studies are performed to show the effectiveness of the proposed model and the method of reducing the number of buses considered for virtual bidding. The impacts of the transmission limits, wind unit location, risk aversion parameters, wind power volatility, and wind and virtual capacities on the price-maker trading strategy are also studied through case studies.

Overbooking for physical examination considering late cancellation and set-resource relationship

BackgroundLate cancellations of physical examination has severe impact on the operations of a physical examination center since it is often too late to fill vacancy. A booking control policy that considers overbooking is then one natural solution. Unlike appointment scheduling problems for clinics and hospitals, in which treating a patient mostly requires only one type of resource, a physical examination set typically requires multiple types of resources. Traditional methods that do not consider set-resource relationship thus may be inapplicable.MethodsWe formulate a stochastic mathematical programming model that maximizes the expected net reward, which is the examination revenue minus overage cost. A complete search algorithm and a greedy search algorithm are designed to search for optimal booking limits for all examination sets. To estimate the late cancellation probability for each individual consumer, we apply logistic regression to identify significant factors affecting the probability. After clustering is used to estimate individual probabilities, Monte Carlo simulation is conducted to generate probability distributions for the number of consumers without late cancellations. A discrete-event simulation is performance to evaluate the effectiveness of our proposed solution.ResultsWe collaborate with a leading physical examination center to collect real data to evaluate our proposed overbooking policies. We show that the proposed overbooking policy may significantly increase the expected net reward. Our simulation results also help us understand the impact of overbooking on the expected number of customers and expected overage. A sensitivity analysis is conducted to demonstrate that the benefit of overbooking is insensitive to the accuracy of cost estimation. A Pareto efficiency analysis gives practitioners suggestions regarding policy determination considering multiple performance indications.ConclusionsOur proposed overbooking policies may greatly enhance the overall performance of a physical examination center.

Simultaneous schedule of trains and track maintenance according to stochastic blockage time

ABSTRACT Regarding the railway infrastructures, planning for track maintenance is a challenging task given the coordination required between train traffic and maintenance operations. In this paper, using stochastic mathematical programming, we have investigated the simultaneous scheduling of trains and operations for a single-track line to minimise both the travel time of trains and maintenance duration. The uncertainty in the blockage duration of the track is considered in the model to reduce the impact of unexpected delays in maintenance operations. The proposed model also considers the practical aspects, which have been less addressed in previous studies regarding track maintenance. We reformulated the stochastic problem as its deterministic equivalent to facilitate solutions for realistic sizes. The computational result obtained for a real-world track shows that the model can efficiently find simultaneous scheduling of trains and operations and suggest optimal blockage duration at different confidence levels. The impact of timetable comparison on the blockage duration for a heavily utilised line has also been evaluated.

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

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

Optimization Model for Integrated Municipal Solid Waste System Using Stochastic Chance-Constraint Programming under Uncertainty: A Case Study in Qazvin, Iran

Municipal solid waste management (MSW) is a factor that affects environmental pollution and the spread of diseases in cities. Therefore, an efficient MSW management system results in reducing the cost of environmental impact by tackling the processes of waste collection, recycling, and disposal. In this study, a biobjective optimization model is developed which aims to minimize the costs of facility location and transportation planning and the emission of environmental pollutants. Furthermore, to consider the uncertain nature of the problem, demand or the volume of the generated waste is considered as a random parameter. As a result, a stochastic mathematical programming model with probable constraints is developed. To solve and validate the model, the ε-constraint approach has been employed. Moreover, for a real-world application of the proposed model, a case study is implemented in Qazvin, Iran. Finally, various problems are solved for different levels of reliability and an efficient MSW system is designed for each of them. Results show that the proposed method was able to achieve Pareto solutions where managers can decide to choose one of them based on their priorities in comparison with the current status. Moreover, results revealed cost and emission would be reduced by increasing confidence level. Finally, a comparison is made between our proposed ε-constraint method and one of the recently used solution approaches.

Stochastic mathematical programs with probabilistic complementarity constraints: SAA and distributionally robust approaches

Stochastic mathematical programs with probabilistic complementarity constraints: SAA and distributionally robust approaches

A multi-objective stochastic programming model for post-disaster management

This paper develops a mathematical model for post-disaster planning with human casualties, which can be considered as operational guidance for the proper use of emergency resources. For this purpose, a stochastic mixed-integer programming model is provided to formulate the problem. The objective functions of the model are (1) maximizing the survival probability of patients, (2) minimizing the maximum of completion time of treatment of all patients, and (3) minimizing the total cost of operations. The model is solved with the ϵ-constraint method. Due to the NP-hardness of the problem which is a significant challenge in the literature, two innovative meta-heuristic algorithms are proposed, i.e. a non-dominated sorting genetic algorithm (NSGA-II) and a multi-objective simulated annealing (MOSA). Finally, a comprehensive computational analysis is performed for evaluation purposes. Also, a case study is made on the earthquake in Iran, which illustrates the real-world application of the model. Highlights A stochastic multi-objective mathematical programming model to allocate patients to hospitals and treat them. NSGA-II and MOSA are proposed, in addition to ϵ-constraint method as solution methods. Performance of two meta-heuristic algorithms is measured with five evaluation metrics. The study showed that NSGA-II is more effective than MOSA. The model was implemented on a real-world case, Tabriz earthquake in Iran.

Simultaneous production scheduling and transportation optimization from mines to port under uncertain material supply

Industrial mining complexes can be optimized using simultaneous stochastic optimization (SSO), which manages the risks associated with meeting production targets while capitalizing on the synergies that exist between the various components of the related mineral value chain. This paper introduces an extension of past SSO approaches for the long-term, allowing to simultaneously optimize the schedule of the production and the mines-to-port transportation of mining complexes under uncertain material supply. The inclusion of mine-to-port transportation scheduling facilitates the analysis of the mines-to-port equipment usage, while generating suitable mine production schedules. The proposed stochastic mathematical program formulation is applied to a two-mine, single-port iron ore mining complex. In doing so, it is shown that the related model is capable of producing optimal production schedules, minimizing deviations from products requirements, and delineating the yearly use of the mine-to-port transportation equipment.