Stochastic Robust Optimization Research Articles

Robust stochastic optimization of needle configurations for robotic HDR prostate brachytherapy.

Ideally, inverse planning for HDR brachytherapy (BT) should include the pose of the needles which define the trajectory of the source. This would be particularly interesting when considering the additional freedom and accuracy in needle pose which robotic needle placement enables. However, needle insertion typically leads to tissue deformation, resulting in uncertainty regarding the actual pose of the needles with respect to thetissue. To efficiently address uncertainty during inverse planning for HDR BT in order to robustly optimize the pose of the needles before insertion, that is, to facilitate path planning for robotic needleplacement. We use a form of stochastic linear programming to model the inverse treatment planning problem. To account for uncertainty, we consider random tissue displacements at the needle tip to simulate tissue deformation. Conventionally for stochastic linear programming, each simulated deformation is reflected by an addition to the linear programming problem which increases problem size and computational complexity substantially and leads to impractical runtime. We propose two efficient approaches for stochastic linear programming. First, we consider averaging dose coefficients to reduce the problem size. Second, we study weighting of the slack variables of an adjusted linear problem to approximate the full stochastic linear program. We compare different approaches to optimize the needle configurations and evaluate their robustness with respect to different amounts of tissuedeformation. Our results illustrate that stochastic planning can improve the robustness of the treatment with respect to deformation. The proposed approaches approximating stochastic linear programming better conform to the tissue deformation compared to conventional linear programming. They show good correlation with the plans computed after deformation while reducing the runtime by two orders of magnitude compared to the complete stochastic linear program. Robust optimization of needle configurations takes on average 59.42s. Skew needle configurations lead to mean coverage improvements compared to parallel needles from 0.39 to 2.94 percentage points, when 8mm tissue deformation is considered. Considering tissue deformations from 4 to 10mm during planning with weighted stochastic optimization and skew needles generally results in improved mean coverage from 1.77 to 4.21 percentagepoints. We show that efficient stochastic optimization allows selecting needle configurations which are more robust with respect to potentially negative effects of target deformation and displacement on the achievable prescription dose coverage. The approach facilitates robust path planning for robotic needleplacement.

Designing a sustainable disruption-oriented supply chain under joint pricing and resiliency considerations: A case study

Supply chains must adopt an appropriate mechanism to deal with interrelated risks. Sustainable development's economic, environmental and social aspects are critical concerns of supply chains. This study presents a three-objective model to design a forward supply chain network considering interrelated operational and disruptive risks. Several mitigative and preventive resilience strategies are implemented to cope with mentioned risks. Also, the optimal prices for products and parts are determined via a joint pricing strategy to enhance supply chain's profitability in disruption incidence. A stochastic robust optimization method is employed to deal with parameters’ uncertainty. The proposed model is applied to an industry of manufacturing chemical processing equipment to validate the formulation and derive practical insights. Results revealed that simultaneously managing disruption risks and uncertainties can improve sustainability goals and reduce the costs associated with risk exposure and protection in the under-study supply chain.

Data-driven stochastic robust optimization of sustainable utility system

Sustainable utility systems reduce reliance on fossil fuels by using renewable energy sources. Multi-scale uncertainties associated with renewable energy and utility systems pose challenges to the modeling and optimization of sustainable utility systems. This study proposes a sustainable retrofit framework for utility systems based on a data-driven stochastic robust optimization approach. Kernel density estimation and fuzzy clustering were employed to capture the uncertainty features in a holistic framework. A life cycle assessment approach was used to calculate the global warming potential (GWP) of the sustainable utility system, and a multi-objective environmental and economic optimization model was developed. A nested decomposition-based algorithm was proposed to solve a large-scale mixed-integer nonlinear programming problem. Finally, a case study of an industrial utility system was conducted to demonstrate the effectiveness of the proposed method. The optimization results show that the proposed method reduces GWP by 9 % after introducing renewable energy and achieves a balance between economic and environmental performance.

Antifragile, sustainable, and agile supply chain network design by considering resiliency, robustness, risk, and environmental requirements.

This research suggests an Antifragile, Sustainable and Agile Supply Chain Network Design (ASASCND) as a new network design that integrates these concepts considering resiliency, robustness, risk, and environmental requirements. The cost function combines a novel method with robust stochastic optimization and Entropic Value atRisk (EVaR). This model combines expected value, maximum and EVaR of cost as an objective function. This research adds antifragility by the effect of learning on variable parameters, sustainability by considering the environmental and social issues, resiliency and agility by flexible capacity, and multi-resource and demand satisfaction constraints to the model. The case study is in the automotive industry. This model compares the main problem by considering antifragility without thinking about antifragility. The ASASCND cost is - 0.3% less than without considering antifragility. In addition, when the conservatism coefficient grows, the cost function increase. In addition, the antifragility coefficient and the confidence level affect positively, and the agility coefficient negatively affects the cost function. Expanding the model scale changes the cost function and time computation because the antifragility coefficient changes variable cost. Finally, managerial insights and practical implications are explained.

Stochastic-robust planning optimization method based on tracking-economy extreme scenario tradeoff for CCHP multi-energy system

The load tracking performance of combined cooling, heating, and power multi-energy system (CCHP-MES) greatly depends on the equipment capacity configuration. And the frequent fluctuations in the source-load uncertainty puts higher demands on the load tracking ability of CCHP-MES. For this reason, this paper proposes a novel tracking-economy extreme scenario tradeoff model by comparing the differences between tracking ability and economic benefit. The corresponding extreme scenarios based on these two objectives are characterized and the concept of uncertainty adjustment parameter is introduced to weigh the preference of extreme scenarios. On this basis, a multi-objective two-stage stochastic-robust optimization approach is developed, where the first stage optimizes the investment cost and the second stage optimizes the operating cost. Case study using typical load demand and renewable energy data in Nanjing verify the efficacy of the optimization model. The results indicate that the proposed tracking-economy extreme scenario can better characterize the frequent fluctuations of uncertainties compared to the traditional definition of extreme scenario. The equipment capacity configuration obtained under tracking-economy extreme scenario tradeoff model can better cope with the uncertainty fluctuations under extreme weather. Depending on different optimization preference, the economic robustness and tracking robustness can be improved by 13.80% and 36.06%, respectively.

A Novel Chaotic Elite Adaptive Genetic Algorithm for Task Allocation of Intelligent Unmanned Wireless Sensor Networks

In recent times, the progress of Intelligent Unmanned Wireless Sensor Networks (IUWSNs) has inspired scientists to develop inventive task allocation algorithms. These efficient techniques serve as robust stochastic optimization methods, aimed at maximizing revenue for the network’s objectives. However, with the increase in sensor numbers, the computation time for addressing the challenge grows exponentially. To tackle the task allocation issue in IUWSNs, this paper introduces a novel approach: the Chaotic Elite Adaptive Genetic Algorithm (CEAGA). The optimization problem is formulated as an NP-complete integer programming challenge. Innovative elite and chaotic operators have been devised to expedite convergence and unveil the overall optimal solution. By merging the strengths of genetic algorithms with these new elite and chaotic operators, the CEAGA optimizes task allocation in IUWSNs. Through simulation experiments, we compare the CEAGA with other methods—Hybrid Genetic Algorithm (HGA), Multi-objective Binary Particle Swarm Optimization (MBPSO), and Improved Simulated Annealing (ISA)—in terms of task allocation performance. The results compellingly demonstrate that the CEAGA outperforms the other approaches in network revenue terms.

Data-driven source-load robust optimal scheduling of integrated energy production unit including hydrogen energy coupling

A robust low-carbon economic optimal scheduling method that considers source-load uncertainty and hydrogen energy utilization is developed. The proposed method overcomes the challenge of source-load random fluctuations in integrated energy systems (IESs) in the operation scheduling problem of integrated energy production units (IEPUs). First, to solve the problem of inaccurate prediction of renewable energy output, an improved robust kernel density estimation method is proposed to construct a data-driven uncertainty output set of renewable energy sources statistically and build a typical scenario of load uncertainty using stochastic scenario reduction. Subsequently, to resolve the problem of insufficient utilization of hydrogen energy in existing IEPUs, a robust low-carbon economic optimal scheduling model of the source-load interaction of an IES with a hydrogen energy system is established. The system considers the further utilization of energy using hydrogen energy coupling equipment (such as hydrogen storage devices and fuel cells) and the comprehensive demand response of load-side schedulable resources. The simulation results show that the proposed robust stochastic optimization model driven by data can effectively reduce carbon dioxide emissions, improve the source-load interaction of the IES, realize the efficient use of hydrogen energy, and improve system robustness.

Optimal scheduling of ethylene plants under uncertainty: An unsupervised learning-based data-driven strategy

Ethylene cracking furnaces are core units that determine the yield of key products, and its cycle scheduling optimization is of great significance to improve the efficiency of the whole plant. However, uncertainties in cracking batch durations may lead to sub-optimal or ineffective scheduling arrangements. This paper develops a novel data-driven stochastic robust optimization strategy to achieve the optimal scheduling of cracking furnaces under uncertainty. The proposed stochastic robust optimization model applies machine learning methods to extract data information and is represented as a bilevel problem: the outer layer is a two-stage stochastic programming, where typical scenarios are clustered by a three-level data mining approach; the inner layer is robust optimization, whose uncertainty set is constructed by an efficient kernel-based approach. An actual ethylene cracking system is investigated and the results show that the proposed method achieves the highest daily profit, performing well in data coverage (92.3%) and change rate (0.826%). Moreover, the sensitivity analysis for the number of clusters and regularization parameter values is performed.

A Robust Stochastic Programming Model for the Well Location Problem: The Case of The Brazilian Northeast Region

Slow-onset disasters, such as drought, are usually more destructive in the long term since they affect the productive capacity of a community, thereby preventing it from recovering using its resources. This requires the leaders and planners of drought areas to establish the best strategies for effective drought management. In this direction, the present work develops a robust stochastic programming approach for the problem of locating artesian wells for the relief of drought-affected populations under uncertainty. Our model considers different demand scenarios and proposes a novel perspective which considers both social and hydrogeological aspects for the location choice, aiming to maximize the affected area’s satisfaction through its prioritization using a composite drought risk index as well as to maximize the probability of success in water prospecting. We present a case study of our robust stochastic optimization approach for the Brazilian Semiarid Region using demand points from the database of Operação Carro-Pipa. Our findings show that a robust solution has a better expected value for the objective function considering all scenarios, so it can help decision makers to plan facility location and demand allocation under demand uncertainty, pointing out the best solution according to their degree of risk aversion.

Disruption-resilient supply chain entities with decentralized robust-stochastic capacity planning

This paper presents decentralized capacity planning models for four types of supply chain entities, classified based on functional similarities, to enhance their resilience. Novel resilience metrics are devised to quantify the proximity of backed-up, fortified, and recovered capacities to the amount of disrupted capacity by analyzing post-disruption events (i.e., absorption, adaptation, and restoration). Based on these metrics, bi-objective optimization models are developed to maximize resilience and cost-efficiency simultaneously to select proper business continuity plans contributing to supply chain entities’ proactive and reactive resilience. Finally, a robust-stochastic optimization method is tailored to address the uncertainties associated with the available time to recover and the impact and occurrence of disruptions. Also, disruption scenarios are simulated using a discrete-time Markov chain. Computational tests confirm the proposed models’ robustness, validity, and generality. The results describe essential features of business continuity plans, prescribe optimal decisions to enhance resilience, and predict how sensitive the resiliency and the imposed costs are to disruptions or proactive/reactive planning.

Microgrids and distribution system resilience assessment: A multi-objective robust-stochastic optimization approach considering social behavior using a deep learning modeling

In this paper, a new multi-objective two-stage robust-stochastic (MOTSRS) optimization approach for assessing microgrids and distribution system resilience is proposed. A novel approach based on deep learning is presented to model social behavior, specifically Social Reactions to Disasters (SRDs), in assessing microgrid and distribution system resilience. It is initially demonstrated that the proposed robust-stochastic approach is efficient for assessing the resilience of microgrids and distribution systems. A new resilience criterion is also proposed to be used for sensitivity analyses at the conservative level of the model. Subsequently, the effect of SRDs on system resilience is investigated through a supervised deep regression method. First, a large dataset is generated through a proposed nested Monte Carlo simulation. A numerical feature and five binary social behavior features are included in the dataset. Next, a feedforward neural network is utilized to predict the load during both normal and disaster circumstances based on the six features. The proposed supervised deep regression model with mean squared errors between 2e-5 and 2e-6 is capable of predicting electrical load during normal and disaster events. It is concluded that the resilience metrics of MGs, as well as expected operation costs and regret values, are influenced by the SRD effect.

A data-driven model for sustainable and resilient supplier selection and order allocation problem in a responsive supply chain: A case study of healthcare system

This research attempts to study the Supplier Selection and Order Allocation Problem (SSOAP) considering three crucial concepts, namely responsiveness, sustainability, and resilience. To do so, the current research develops a Multi-Stage Decision-Making Framework (MSDMF) to select potential suppliers and determine the quantity of orders. The first stage aims at computing the scores of the suppliers based on several indicators. To do this, a novel decision-making approach named the Stochastic Fuzzy Best–Worst Method (SFBWM) is developed. Then, in the second stage, a Multi-Objective Model (MOM) is suggested to deal with supplier selection and order allocation decisions. In the next step, a data-driven Fuzzy Robust Stochastic (FRS) optimization approach, based on the fuzzy robust stochastic method and the Seasonal Autoregressive Integrated Moving Average (SARIMA) methods, is employed to efficiently treat the hybrid uncertainty of the problem. Afterwards, a novel solution method named the developed Chebyshev Multi-Choice Goal Programming with Utility Function (CMCGP-UF) is developed to obtain the optimal solution. Moreover, given the crucial role of the Medical Equipment (ME) industry in society’s health, especially during the recent Coronavirus disease, this important industry is taken into account. The outcomes of the first stage demonstrate that agility, cost, GHG emission, quality, robustness, and Waste Management (WM), respectively, are the most important criteria. The outcomes of the second stage determine the selected suppliers, utilized transportation systems, and established sites. It is also revealed that demand directly affects all the objective functions while increasing the rate of disruptions has a negative effect on the sustainability measures.

Practicable robust stochastic optimization under divergence measures with an application to equitable humanitarian response planning.

We seek to provide practicable approximations of the two-stage robust stochastic optimization model when its ambiguity set is constructed with an f-divergence radius. These models are known to be numerically challenging to various degrees, depending on the choice of the f-divergence function. The numerical challenges are even more pronounced under mixed-integer first-stage decisions. In this paper, we propose novel divergence functions that produce practicable robust counterparts, while maintaining versatility in modeling diverse ambiguity aversions. Our functions yield robust counterparts that have comparable numerical difficulties to their nominal problems. We also propose ways to use our divergences to mimic existing f-divergences without affecting the practicability. We implement our models in a realistic location-allocation model for humanitarian operations in Brazil. Our humanitarian model optimizes an effectiveness-equity trade-off, defined with a new utility function and a Gini mean difference coefficient. With the case study, we showcase (1) the significant improvement in practicability of the robust stochastic optimization counterparts with our proposed divergence functions compared to existing f-divergences, (2) the greater equity of humanitarian response that the objective function enforces and (3) the greater robustness to variations in probability estimations of the resulting plans when ambiguity is considered.

Robust Multi-objective optimal dispatching model for a novel island micro energy grid incorporating biomass waste energy conversion system, desalination and power-to-hydrogen devices

To realize renewable and self-sustainable energy supply in island region, based on geographical characteristics with abundant renewable resources, an optimal model for island micro energy grid (MEG) is designed incorporating biomass waste energy conversion system (ECS), desalination, and power-to-hydrogen (BSP-MEG) Firstly, the mathematical model is designed, including models of power generators, ECS, desalination and power-to-hydrogen (P2H) devices, etc. Next, the multi-objective scheduling optimization model is designed, containing conventional scheduling model (Scheduling optimization objectives and constraints established with minimum operation and environment costs) and stochastic scheduling model (Minimum Conditional Value-at-Risk objective specific to volatility and uncertainty of renewable generations based on robust stochastic optimization method). Then, to solve the multi-objective optimization problem (MOP), a hybrid differential evolution algorithm is proposed based on local optimal and external archiving strategies. Finally, the MEG of YongXing Island is selected as an example. The results show (1) BSP-MEG effectively realized multi-energy cooperative optimization, and promote intra-day peak shaving. (2) BSP-MEG reduced operating costs, environmental costs and Conditional Value-at-Risk (CVaR) by 78.2%, 61.8% and 77.9% respectively, while curtailment rate by 25.6 to 0.9%. (3) Whether in general scenario or worst, BSP-MEG can realize self-production and self-sale of energy and material, of which risk resistance ability is better. (4) By designing local optimal and external archiving strategies, hybrid differential evolution algorithm performs better in solving complex MOP. In general, the optimization model proposed in this paper can improve the utilization of renewable resources, alleviate the shortage of fresh water, and help realize renewable and sustainable energy supply.

Online-to-offline on the railway: Optimization of on-demand meal ordering on high-speed railway

On-demand meal ordering on high-speed railway is an emerging on-demand service that is becoming increasingly popular. One key challenge about this service is how to design the ordering system to maximize profits while guaranteeing timely delivery. However, few related studies have been done for this new business. In this paper, we first develop a mathematical model to optimize the existing ordering procedure, which sets a deadline for order placement. Furthermore, we propose a pre-ordering policy which cancels the order deadline but ensures timely delivery via stocking up. Considering the ambiguity in demand distributions and uncertainty in delivery time, we develop distributionally robust stochastic optimization models for both the existing and the pre-ordering procedures. Cutting-plane algorithms are developed to solve the proposed models. Computational studies on a real-world high-speed railway line in China show that our proposed new policy could simultaneously increase service availability, improve the profit, and reduce delay rate.

Urban integrated energy system stochastic robust optimization scheduling under multiple uncertainties

The uncertainties of the wind power generation, solar power generation, electricity load, and heat load bring difficulties to the optimal scheduling of the urban integrated energy system (UIES), which influence its secure and reliable operation. To promote the utilization of renewable energy and enhance the scheduling capacity of UIES, a two-stage stochastic robust optimization model with the day-ahead market and the real-time market is developed. The goal is to optimize the UIES’s profit in the day-ahead market and minimize the total system cost in the real-time market, accounting for the most unfavorable situations of uncertainties. To address multiple uncertainties, a four-dimensional ambiguity set considering multiple uncertainties is constructed by using the Dirichlet process mixture model (DPMM) and the variational inference algorithm. Meanwhile, an accelerated column-and-constraint generation algorithm (AC&CG) is presented for iterative solution of the proposed model, with the second stage reformulated via Lagrange dual theory. Besides, a novel acceleration strategy is adopted, which contributes the algorithm to converging faster which can also improve the iteration speed. Finally, simulation examples are used to demonstrate the effectiveness of the proposed model in comparison to stochastic programming and robust optimization. This paper may provide theoretical guidance for relevant research and application.

Feedback-Optimizing Heuristic Model Predictive Control Applied to a Wind Turbine for Suppressing the Tower Vibrations

To have accurate control of a wind turbine, especially when facing wind loads, it is vital to model the dynamic behavior of the system accurately and select the controller parameters properly. To mitigate these challenges, here, an optimal model-based controller is utilized using particle swarm optimization to suppress the vibrations of the tower for an NREL 5 MW turbine in the presence of the turbulence wind. For this purpose, first, a discrete state-space model is derived for the wind turbine based on the finite difference method, and the validity of the model structure is verified with its analytical solution. Then, the structure of a model predictive controller is implemented on the model of the wind turbine with optimization of the plant behavior, extracted from state-space form, in a prediction horizon. Next, to prevent the dependence of the proposed controller structure on its parameters, a number of training scenarios including the turbulent wind with different intensities are run in parallel. The cost of each scenario is continually updated based on the tracking error, the control effort and the control command generation time in total run time. Consequently, the best parameters of the controller are adjusted using a stochastic robust optimization algorithm such as particle swarm optimization. Results demonstrate that the proposed approach has an excellent performance in dissipating the tower vibration.

A stochastic-robust optimization model for inter-regional power system planning

This study develops a stochastic-robust optimization model for inter-regional power system planning. The model is formulated as Mathematical Programs with Equilibrium Constraints (MPEC) with two levels of decision makers. In the bottom-level, various individual regions separately make generator investment, operating, and trading plans for the minimization of their own costs in a perfectly competitive market. In the top-level, a central system operator makes the investment plan of cross-border transmission lines for the minimization of total system cost, anticipating how generators in various regions respond to those investments. Two different levels of data uncertainties are considered in the problem, namely scenario and local uncertainties. The uncertainties are handled by combining the advantages of stochastic programming and robust optimization methods. The proposed model is applied for power system planning in the Association of Southeast Asian Nations (ASEAN) between 2020 and 2040. The modelling results show that the integration of a cross-border power grid results in a substantial shift in the generation portfolio, and can potentially reduce total costs by up to 6.0%. The cost savings primarily result from the increased utilization of renewable energy resources, facilitated by cross-border electricity transmission. We assessed the performance of our proposed approach in handling data uncertainties by comparing it with existing practices. The comparison results demonstrate that our stochastic-robust approach not only enhances the capacity expansion plan’s robustness to meet power demand and CO2 emissions targets, but also significantly reduces investment and operational costs.

A Simulation-Based Optimization Model for Control of Soil Salinization in the Hetao Irrigation District, Northwest China

The average annual water diversion of the Hetao Irrigation District (HID) from the Yellow River is 4.5 billion cubic meters, mainly used for surface irrigation. Because the groundwater depth is shallow, strong evaporation conditions and unmatched irrigation conditions lead to serious soil salinization in the area; thus, the irrigation area’s ecological environment is fragile. Based on the current situation of the Yellow River irrigation project in the area, an interval two-stage robust stochastic optimization model is proposed to address the problem. In 2015, the Shuguang Experimental Station in the middle of the HID, Inner Mongolia, discussed the impact of different degrees of water–salt coordinated regulation on water consumption, yield and price of wheat, maize and sunflower under drip irrigation conditions. The obtained results provide the water shortage and water distribution targets of multiple water sources and multiple water levels in five irrigation areas of the HID. Those water distribution targets were used as the main input parameter and entered into the SALTMOD model based on the principle of water and salt balance. The output included data on groundwater mineralization and groundwater depth. It was observed that (1) integrated interval two-stage robust stochastic programming and the SALTMOD Model to couple optimization model under uncertainty can simulate a model together; (2) systemic risk issues were considered; and (3) the proposed method can be applied to the HID in northwest China to solve the soil salinization control problem. This approach is applicable to arid and semiarid regions that face similar problems.

Dynamic pricing and inventory control for multiple products in a retail chain

The biggest problems in the retail industry are related to price allocation to keep the product attractive to the market, without allowing arbitrage, together with adequate inventory control involving substitution and complementarity. The above objectives gains relevance if the uncertain behavior of the market is taken into account. Most of the existing literature addresses each of the above problems independently or as a combination of some of them and not as a whole that is related to each other. In this research, we present a model that allows setting optimal dynamic pricing and inventory policies in each period in which a planning horizon is divided. When assigning the price, the model avoids the presence of arbitrage, by allowing the use of fixed price policies (same price for the product in the stores) and/or variable price policies (different prices for the product in the stores). When making replenishments, it allows having an adequate quantity of each product, since it incorporates substitution and complementarity between them. The market demand in each store is represented by a probability distribution that is a function of the market by means of a seasonality parameter, price sensitivity, price and inventory, subject to a set of restrictions. We find that demand has uncertainty in the seasonality parameter and in the price sensitivity, due to variations outside the historical data with which it is calculated. To handle this uncertainty, we use a Robust Stochastic Optimization formulation, which we solve in an approximate way, using a scenario generation technique. We show computational experiments applied to a case with industrial data. The results show the advantages of using this methodology, allowing managers to make the right decisions and thus improve operating results.