Robust Optimization Methodology Research Articles

Robust optimization of hypoid gear contact performance considering tooth form error: Design sensitivity and Pareto front

Optimization of hypoid gear contact performance is challenging due to its nonconvex nature and the handling of non-linear constraints involving complex gear contact behavior. In addition, the contact performance in the existing optimization schemes is highly reliant on the manufacturing quality, for example, the uncertain tooth form errors induced by machining tolerances and blade wear are usually ignored. To tackle these problems, a robust optimization methodology is proposed for hypoid gear contact performance. A finite element/contact mechanics model is established, which considers the stochastic tooth form errors following a Gaussian distribution model. The non-linear constraints are incorporated to avoid edge contact under loaded conditions, and tooth surface interference due to error. This methodology guarantees the feasible solution space and reduces the optimization complexity by the independent identification model. The identification model of the optimal solution is flexible in the parameter selection while keeping tooth depth. This non-deterministic problem is optimized based on the pattern search method. The Pareto front and design sensitivity analysis demonstrate the effectiveness and robustness of the proposed optimization methodology.

Robust Optimization Approaches in Inventory Management: Part A - The Survey

This work, the first part (Part A) of a comprehensive study, presents a survey on Robust Optimization (RO) in inventory management, highlighting its role in addressing uncertainties. This survey reviews various modeling frameworks, types of uncertainties, decision-making criteria, and decision rules essential to RO. The subsequent part (Part B) offers a comparative study, analyzing robust inventory models and highlighting key analytical and numerical contributions. This survey critically evaluates the effectiveness of RO in managing model uncertainty, enhancing decision-making processes, deriving structural insights, and boosting computational efficiency. Additionally, it discusses the limitations and challenges of applying RO to inventory management. While acknowledging the foundational role of traditional inventory literature in establishing essential theories, optimal policies, and efficient algorithms, this paper addresses a significant gap by focusing on inventory management through the lens of RO methodology. Together with the comparative study, these works encapsulate the current state of robust inventory management, shedding light on future research directions and ongoing challenges in this evolving field.

Robust radiotherapy planning with spatially-based uncertainty sets

Radiotherapy treatment planning is a challenging large-scale optimization problem plagued by uncertainty. Following the robust optimization methodology, we propose a novel, spatially based uncertainty set for robust modeling of radiotherapy planning, producing solutions that are immune to unexpected changes in biological conditions. Our proposed uncertainty set realistically captures biological radiosensitivity patterns that are observed using recent advances in imaging, while its parameters can be personalized for individual patients. We exploit the structure of this set to devise a compact reformulation of the robust model. We develop a row-generation scheme to solve real, large-scale instances of the robust model. This method is then extended to a relaxation-based scheme for enforcing challenging, yet clinically important, dose–volume cardinality constraints. The computational performance of our algorithms, as well as the quality and robustness of the computed treatment plans, are demonstrated on simulated and real imaging data. Based on accepted performance measures, such as minimal target dose and homogeneity, these examples demonstrate that the spatially robust model achieves almost the same performance as the nominal model in the nominal scenario, and otherwise, the spatial model outperforms both the nominal and the box-uncertainty models.

Stochastic Optimization Approaches for an Operating Room and Anesthesiologist Scheduling Problem

Improving Operating Room, Surgery, and Anesthesiologist Scheduling Using Stochastic and Robust Optimization Efficient planning and scheduling of operating room (OR) activities is crucial for managing costs and delivering high-quality surgical care. However, this task is extremely complex for several reasons. First, it requires coordinating multiple resources, such as ORs and anesthesiologists. Second, in addition to limited OR capacity and time, there is a significant shortage of anesthesiologists required to perform surgeries. Third, surgery durations are uncertain and difficult to predict. Ignoring such uncertainty may lead to substantial overtime, idling, and surgery delays, among other schedule deficiencies. Thus, hospital managers could benefit greatly from advanced methodologies to improve OR utilization, surgical care, and quality as well as to minimize OR operational costs. In “Stochastic Optimization Approaches for an Operating Room and Anesthesiologist Scheduling Problem,” M. Y. Tsang, K. S. Shehadeh, F. E. Curtis, B.R. Hochman, and T. E. Brentjens propose computationally tractable stochastic programming and distributionally robust optimization methodologies for an integrated allocation, assignment, sequencing, and scheduling problem under uncertainty involving multiple ORs, anesthesiologists, and surgery types. Using real-world surgery data and a case study from a health system in New York, they conduct extensive experiments demonstrating the computational efficiency of the proposed methodologies, allowing for their implementation in practice. Moreover, they show the negative consequences of adopting the existing non-integrated approaches and provide valuable practical insights.

Optimizing the Value of a CO2 Water-Alternating-Gas Injection Project under Geological and Economic Uncertainties

Summary The numerical simulations required for the robust optimization (RO) of the alternating injection of water and CO2 in hydrocarbon reservoirs are computationally expensive due to engineering, geological, and economic uncertainties. Using approximation models of the desired objective function(s) can significantly decrease the cost associated with the optimization routines while providing an adequate sampling of the input variables or uncertainties. In this study, we optimized the value of a water-alternating-gas (WAG) injection project in a Niger-Delta oil reservoir by applying the Markowitz classical theory to a suitable approximation model of the objective function. Our RO methodology incorporated significant geological and economic uncertainties—such as uncertainties due to the upscaling of the coarse-scale reservoir model and those due to the lack of other relevant geological and economic data—to the optimization routine to create better operating strategies for the projects that are risk-quantified. To compute the objective function, a novel economic model for the CO2 sequestration processes in the Niger-Delta hydrocarbon basin was applied. The result obtained in the presented case showed that a net present value (NPV) of at least USD 65.98 million was derived from the project depending on the engineer’s or user’s confidence level. Overall, the applicability of proxy models to RO routines was demonstrated. As a result, investment decisions that accounted for uncertainties and any potential variation in the field’s operation and development were made.

Decarbonizing OCP

Problem definition: We present our collaboration with the OCP Group, one of the world’s largest producers of phosphate and phosphate-based products, in support of a green initiative designed to reduce OCP’s carbon emissions significantly. We study the problem of decarbonizing OCP’s electricity supply by installing a mixture of solar panels and batteries to minimize its time-discounted investment cost, plus the cost of satisfying its remaining demand via the Moroccan national grid. OCP is currently designing its renewable investment strategy, using insights gleaned from our optimization model, and has pledged to invest 130 billion Moroccan dirham (MAD) (approximately 13 billion U.S. dollars (USD)) in a green initiative by 2027, a subset of which involves decarbonization. Methodology/results: We immunize our model against deviations between forecast and realized solar generation output via a combination of robust and distributionally robust optimization. To account for variability in daily solar generation, we propose a data-driven robust optimization approach that prevents excessive conservatism by averaging across uncertainty sets. To protect against variability in seasonal weather patterns induced by climate change, we invoke distributionally robust optimization techniques. Under a 10 billion MAD (approximately 1 billion USD) investment by OCP, the proposed methodology reduces the carbon emissions that arise from OCP’s energy needs by more than 70%, while generating a net present value (NPV) of 5 billion MAD over a 20-year planning horizon. Moreover, a 20 billion MAD investment induces a 95% reduction in carbon emissions and generates an NPV of around 2 billion MAD. Managerial implications: To fulfill the Paris climate agreement, rapidly decarbonizing the global economy in a financially sustainable fashion is imperative. Accordingly, this work develops a robust optimization methodology that enables OCP to decarbonize at a profit by purchasing solar panels and batteries. Moreover, the methodology could be applied to decarbonize other industrial consumers. Indeed, our approach suggests that decarbonization’s profitability depends on solar capacity factors, energy prices, and borrowing costs. History: This paper has been accepted as part of the 2023 Manufacturing & Service Operations Management Practice-Based Research Competition. Supplemental Material: The e-companion is available at https://doi.org/10.1287/msom.2022.0467 .

İnşaat Maliyet Endeksi Tahmininde Holt-Winters Üstel Düzeltme Parametrelerinin PSO ve İleri Walk-Forward Cross-Validation ile Optimizasyonu

This research aims to enhance the accuracy of Construction Cost Index (CCI) forecasting using Holt-Winters exponential smoothing (ES) by optimizing its parameters, focusing on minimizing the Mean Absolute Percentage Error (MAPE) for precise CCI forecasts. To reach this aim, The Holt-Winters model parameters are optimized through Particle Swarm Optimization (PSO) and Walk-Forward Cross-Validation (WFCV). PSO, a metaheuristic optimization algorithm, is being applied to search for optimal values of the smoothing parameters (alpha, beta, and gamma) that determine the weightage of past observations, trends, and seasonality, respectively. WFCV is assessed the model's performance and ensures robustness. Reduced MAPEs of 22 for CCI forecasts and 2 for training data are the findings of the optimized Holt-Winters model. The obtained alpha, beta, and gamma values are 0.99, 0.77, and 0, respectively, highlighting the importance of while neglecting seasonality. Convergence graphs demonstrate the superiority of the optimization approach over conventional parameter values or random selections. By employing PSO and WFCV, the study efficiently fine-tunes the Holt-Winters model for precise CCI forecasting. Optimized parameter values enable data driven decision-making in construction project cost estimation and budget management. This research contributes a reliable and robust optimization methodology for CCI forecasting, supporting advancements in the field.

Hierarchical multi-scale parametric optimization of deep neural networks

AbstractTraditionally, sensitivity analysis has been utilized to determine the importance of input variables to a deep neural network (DNN). However, the quantification of sensitivity for each neuron in a network presents a significant challenge. In this article, a selective method for calculating neuron sensitivity in layers of neurons concerning network output is proposed. This approach incorporates scaling factors that facilitate the evaluation and comparison of neuron importance. Additionally, a hierarchical multi-scale optimization framework is proposed, where layers with high-importance neurons are selectively optimized. Unlike the traditional backpropagation method that optimizes the whole network at once, this alternative approach focuses on optimizing the more important layers. This paper provides fundamental theoretical analysis and motivating case study results for the proposed neural network treatment. The framework is shown to be effective in network optimization when applied to simulated and UCI Machine Learning Repository datasets. This alternative training generates local minima close to or even better than those obtained with the backpropagation method, utilizing the same starting points for comparative purposes within a multi-start optimization procedure. Moreover, the proposed approach is observed to be more efficient for large-scale DNNs. These results validate the proposed algorithmic framework as a rigorous and robust new optimization methodology for training (fitting) neural networks to input/output data series of any given system. Graphical Abstract

Destination choice modeling with spatially distributed constraints

Destination choice models are a key component of any transport and land-use model. Applications in agent-based models allow for destination choice on an individual level including personal variables, like trip purpose, or situational variables. Commonly applied methodologies stem from econometrics, discrete choice theory and utility maximization using either revealed or stated preference data. This paper presents a framework to integrate cross-section flows between distinct geographic areas, which can be obtained from cordon surveys or mobile phone data. Proposed optimization methodology—based on extended shadow price theory—accommodates these complementary data sources as spatially distributed constraints, in addition to the destination capacity constraints such as workplaces. The new generic and robust optimization methodology accounts for constraints as observed on cross-section flows and destination capacities while maintaining econometric choice model theory. As a proof of concept, the suggested methodology is successfully applied in a real-case, agent-based application covering the tri-national Basel region with about 2 million residents, and a large set of 2 · 104 distinct destination alternatives. Due to different wage levels in all three countries and other reasons, the region’s cross-border commuter flows are highly asymmetric. Including data on cross-border flows obtained from a cordon survey, the choice model’s mean deviation declines by 20% and more on a cross-section level and even more so on a choice alternative level, compared to calculations ignoring shadow prices. Moreover, multiple scenario calculations show considerable improvements in planning and forecasting applications. The results demonstrate the suitability and relevance of the proposed approach to optimize destination choice models with limited destination capacities in geographical regions usually characterized by travel demand asymmetries.

Coupling multi-fidelity xRAGE with machine learning for graded inner shell design optimization in double shell capsules

Bayesian optimization has shown promise for the design optimization of inertial confinement fusion targets. Specifically, in Vazirani et al. [Phys. Plasmas 28, 122709 (2021)], optimal designs for double shell capsules with graded inner shells were identified using one-dimensional xRAGE simulation yield calculations. While the machine learning models were able to accurately learn and predict one-dimensional simulation target performance, using simulations with higher fidelity would improve design optimization and better match with the expected experimental performance. However, higher fidelity physics modeling, i.e., two-dimensional xRAGE simulations, requires significantly larger computational time/cost, usually at least an order of magnitude, in comparison with one-dimensional simulations. This study presents a multi-fidelity Bayesian optimization, in which the machine learning model leverages low-fidelity (one-dimensional xRAGE) and high-fidelity (two-dimensional xRAGE) simulations to more accurately predict “pre-shot” target performance with respect to the expected experimental performance. By building a multi-fidelity Bayesian optimization framework coupled with xRAGE, the low-fidelity and high-fidelity simulations are able to inform one another, such that we have: (1) improved physics modeling in comparison with using low-fidelity simulations alone, (2) reduced computational time/cost in comparison with using high-fidelity simulations alone, and (3) more confidence in the expected performance of optimized targets during real-world experiments. In the future, we plan to use this robust multi-fidelity Bayesian optimization methodology to expedite the design of graded inner shells further and eventually full capsules as a part of the current double shell campaign at the National Ignition Facility.

Multi-objective robust optimization of reservoir operation for real-time flood control under forecasting uncertainty

Flood control operation is one of the effective measures to reduce flood risks. Since flood forecasting plays a critical role in real-time reservoir flood control operation, it is necessary to involve forecasting uncertainty in the optimal reservoir operation to inform decision-making. It has always been a difficult task to reduce flood risks by adjusting flood control strategies. To tackle this challenge, this study developed a multi-objective robust optimization methodology for real-time reservoir flood control operation, which mainly coped with forecast uncertainty. Three machine learning (ML) models, including back propagation neural network (BP), long short-term memory neural network (LSTM) and extreme learning machine (ELM) were adopted to forecast the short-term reservoir inflow. A stacking ensemble multi-ML model (SEM) was then applied to integrate the forecasting results of the above models. Furthermore, a multi-objective robust optimal operation model (MOROU) integrating both upstream and downstream safety was established to reduce flood risks and evaluate the impact of forecasting uncertainty on the reservoir operation. To maximize the efficiency of reservoir utilization, this study defined a new indicator of reservoir reserved capacity adaptation (RRCA) as one of the optimization indicators. To better solve the complex multi-objective problem, a scenario to point (STP) method was proposed for searching robust solutions of multi-objective optimization models. Methodologies were validated through an application to the Lishimen reservoir, China. Three main conclusions were derived from the study: (1) Three ML models performed well in flood forecasting with BP being slightly better than the other two, and the SEM method was found to be able to incorporate the characteristics of each model, outperforming the individual models; (2) MOROU showed a narrower distribution of flood risk both upstream and downstream and achieved an approximately 1.5% reduction in the maximum value of the highest water level; (3) RRCA was verified to enable to reduce the discharge flow by an average of 4.52% without occupying additional flood control reservoir capacity, confirming that it can be used as an optimization indicator to improve the utilization of the reservoir. The proposed method can provide robust strategies for real-time reservoir flood control under forecast uncertainty.

Robust multiobjective scheme for closed-loop supply chains by considering financial criteria and scenarios

This paper considers the closed-loop supply chain design problem by examining financial criteria and uncertainty in the parameters. A robust multiobjective optimization methodology is proposed by considering financial measures such as maximizing the net present value (NPV) and minimizing the financial risk (FR). The proposed methodology integrates various multiobjective optimization elements based on epsilon constraints and robustness measurements through the FePIA (named after the four steps of the procedure: Feature–Perturbation–Impact–Analysis) methodology. Similarly, an analysis of the parameter variability using scenarios was considered. The proposed method's efficiency was tested with real information from a multinational company operating in Colombia. The results show the effectiveness of the methodology in addressing real problems associated with supply chain design.

Fusion of Water Evaporation Optimization and Great Deluge: A Dynamic Approach for Benchmark Function Solving

The Water Evaporation Optimization - Great Deluge explores the synergy between the Water Evaporation Optimization Algorithm (WEOA) and the Great Deluge Algorithm (GDA) to create a novel fusion model. This research investigates the efficacy of combining these two powerful optimization techniques in addressing benchmark problems. The fusion model incorporates WEOA's dynamic exploration-exploitation dynamics and GDA's global search capabilities. By merging their strengths, the fusion model seeks to enhance convergence efficiency and solution quality. The study presents an experimental analysis of the fusion model's performance across a range of benchmark functions, evaluating its ability to escape local optima and converge towards global optima. The results provide insights into the effectiveness of the fusion model and its potential for addressing complex optimization challenges., a comprehensive performance analysis of the application of the proposed fusion model to a curated set of widely acknowledged benchmark functions, renowned for their role in evaluating the capabilities of optimization algorithms, is undertaken. By rigorously evaluating the convergence characteristics, solution quality, and computational efficiency of the algorithm, a thorough understanding of the strengths and limitations of WEOA is aimed to be provided. Through meticulous comparisons with established optimization techniques, illumination of the aptitude of WEOA in addressing diverse optimization challenges across a spectrum of problem landscapes is intended. The analytical insights, not only advancing the understanding of WEOA's applicability, but also furnishing valuable guidance for both researchers and practitioners in search of robust optimization methodologies, are proffered.

A general methodology for reliability-based robust design optimization of computation-intensive engineering problems

Abstract As for complex engineering structures, their single deterministic solutions need to be calculated with the help of computation-intensive finite element software, and the corresponding computation time always lasts very long. When referring to robust design optimization of complex engineering structures, the above expensive job should be repeatedly carried out for a considerable amount of time during the full optimization process. Hence, such repetitive jobs may be difficult to complete for most engineering applications. Aiming at this problem, the paper proposes a general methodology for reliability-based robust design optimization (RBRDO). Firstly, the improved formulation of the RBRDO problem is proposed based on the conventional RBRDO form, and the original probabilistic constraint functions in the improved formulation are changed into the deterministic ones by means of employing the performance measure approach (PMA). Secondly, the above-mentioned improved RBRDO problem is approximately replaced by a new sequence of approximate suboptimizations. For each suboptimization, its original PMA functions are replaced by the approximate explicit form only concerning the deterministic design variables rather than random variables. In this way, each suboptimization is deterministic. Furthermore, the trust-region method is adopted to ensure that the new sequence of suboptimization converge to the original improved RBRDO problem. Lastly, a test problem and three applications are adopted to demonstrate the effectiveness and efficiency of the above-proposed methods. PACS(optional, as per journal): 75.40.-s; 71.20.LP

Robust optimization for lot-sizing problems under yield uncertainty

Production yield can be highly volatile and uncertain, especially in industries where exogenous and environmental factors such as the climate or raw material quality can impact the manufacturing process. Thus, for production planning, it is necessary to take into account the production yield uncertainty to obtain robust and efficient plans. In this paper, we consider lot-sizing problems under yield uncertainty. We propose a multi-period, single-item lot-sizing problem with backorder and yield uncertainty via a robust optimization methodology. First, we formulate a robust model under a budgeted uncertainty set, which is optimized under the worst case perspective to ensure the feasibility of the proposed plan for any realization of the yield described by the uncertainty set. Second, we analyze the structure of the optimal lot-sizing solution, and we derive the optimal robust policy for the special case of the inventory management problem under a box uncertainty. These results help us develop a dynamic program with polynomial complexity for the lot-sizing problem with stationary yield rate. Finally, extensive computational experiments show the robustness and effectiveness of the proposed model through an average and worst case analyses. The results demonstrate that the robust approach immunizes the system against uncertainty. Moreover, a comparison of the robust model with the nominal model, the deterministic model with safety stock, and the stochastic model shows that the robust model balances the costs better by reducing the backorders at the expense of more often producing a larger amount of goods.

A Novel Methodology for Robust Topology Optimization Considering Manufacturing Errors and Topology Deviations

In topology optimization (TO), the finally manufactured topology will inherently deviate from the optimized one due to both curved lines or smoothed surfaces being used in the real-world topology and the manufacture error (ME). Generally, such deviation or tolerance is called an uncertainty in the topology. However, the deviation of the built topology from the optimized one may result in degradation of the performance of the optimized solution for microscale magnetic devices. Therefore, robust optimization (RO) considering ME and topology deviations (TDs) has become a hot concern in TO problems. Consequently, it is demanding to develop feasible and efficient procedures for TOs considering MEs and TDs. In this respect, a novel methodology based on the min-cut theorem and Bezier curve (MCBC) is first proposed. Based on the min-cut-based methodology for deterministic TOs, the Bezier curve is proposed to generate interface distortion realistically. The numerical result on the TO of a magnetic actuator has demonstrated that the proposed methodology has a good ability to obtain optimized topology insensitive to MEs and TDs.

An efficient energy planning model optimizing cost, emission, and social impact with different carbon tax scenarios

This study proposes a modified energy planning model that considers a broad range of future uncertainties. Modifications to hybrid stochastic robust optimization and robust optimization methodology allow for the introduction of multi-objective functions that reflect the various dimensions of energy planning including cost, emission, and social impact. A lexicographical and weighted-sum type multi-objective function are used in formulating the proposed optimization model. Changing the priorities of the objective functions generates different energy policies, which are then compared. Data envelopment analysis is applied to measure the energy efficiency of each optimal energy policy produced by the energy planning model. Energy efficiency is defined as the ability to satisfactorily address five main aspects—cost, emissions, social impact, employment, and security. An updated power development plan for Thailand is used as an illustrative case study. The empirical analysis indicates that a policy prioritizing the environment first, followed by social impact and cost, is the most efficient policy among the five alternatives considered, with an average efficiency score of 0.9988. Sensitivity analysis involving various carbon tax scenarios is used to establish the importance of weight selection in the weighted-sum method. Specifically, the empirical case study reveals that when the weighted-sum method is used, increasing the carbon tax increases the relative policy efficiency score. Results from the case study offer quantitative support for policy makers seeking to devise an efficient energy policy that meets extensive requirements while still dealing with the bounds of uncertain future projections.

Methodology and applications of robust optimization

Methodology and applications of robust optimization

Optimal Operation of Battery Energy Storage Under Uncertainty Using Data-Driven Distributionally Robust Optimization

This paper proposes a conditional value-at-risk (CVaR)-based approach to deal with uncertainties in optimizing the participation of a battery storage in both the electricity market and demand-side management (DSM). A data-driven distributionally robust optimization (DDRO) methodology is used to solve the proposed mean-risk portfolio optimization model. The participation of a battery in DSM along with day-ahead and real-time markets (e.g., energy, spinning reserve, regulation up, and regulation down) faces uncertainties in market-clearing prices, energy demand, and available power capacity. Since in this application, the distribution of the uncertain parameters is only observable through a finite training dataset, the proposed DDRO methodology uses the Wasserstein metric to construct a ball in the space of probability distributions, which is centered at the uniform distribution on the training samples. Then, based on the worst-case distribution within this Wasserstein ball, it seeks decisions that perform best. The proposed DDRO problem over Wasserstein balls is reformulated as a finite convex problem, tested, and verified using real market data and supply/demand profiles. The results are compared with the ones obtained from other optimization methods, including deterministic and classical stochastic optimization, which validates the high performance of the proposed DDRO over finite historical sample data.

Robust optimal design of low frequency metamaterials considering the fabrication imperfections and parameter fluctuations

PurposeThe purpose of this paper is to develop a robust optimization methodology for metamaterial (MM) unit designs to minimize the effect of manufacturing and operational uncertainties.Design/methodology/approachA new robustness quantification function, applicable to both convex and nonconvex relationships between the mean and the standard deviation, is introduced. A distance-based local radial basis function network surrogate model is proposed to substitute the global radial basis function network to reduce the heavy computational cost without any scarification on the solution accuracy.FindingsThe optimized results of a prototype MM unit demonstrate the feasibility and merit of the proposed methodology. The proposed methodology outperforms the existing ones in both performance and robust parameters in the design of a prototype MM unit.Originality/valueIt provides a robust optimization methodology for MM units when considering the imperfections in fabrications and fluctuations in operation and environment conditions in engineering applications.