Multi-objective Robust Optimization Research Articles

Development of a novel multi-fidelity meta modeling approach for robust multi-objective optimization of a natural gas-hydrogen/diesel dual fuel engine

Multi-fidelity modeling (MFM) is an evolving field that matches low-fidelity models (LFM) and high-fidelity models (HFM) to get better solutions with low computational cost. However, improving the duality between accuracy and computational cost remain challenging, particularly for complex problems such as dual fuel engines. This paper contributes to the MF modeling cost-effectiveness improvement by proposing a new approach to solve large-dimensional multi-objective optimization problems. The first step is to build a meta-model based on the LF model, which will be subjected to a comet-governed analysis to detect potential areas where the uncertainty on the LF model is relatively high. Then, a design of experiment (DOE) will be developed based on the results of this analysis to construct an initial HF model. Finally, an iterative loop will be activated to improve the accuracy of the MF model using a well-weighed combination of the details delivered by the LF model correction via the HF model and the HF meta-model. The developed approach is validated on four different mathematical benchmarks with different difficulties, compared with four different MF modeling strategies. This validation shows that the proposed MF modeling is competitive and can produce solutions as accurate as the HF model while reducing significantly the overall computation time by up to 50%. As an engineering application, the operating conditions in a natural gas-hydrogen/diesel dual fuel engine in terms of compression ratio, pilot injection timing, and EGR are optimized. A reduction of 46%, 68%, and 96% was achieved for HC, NOx, and the knocking index, respectively, while an increase in thermal efficiency of about 5% was obtained.

Aggregate production planning and energy supply management in steel industry with an onsite energy generation system: A multi-objective robust optimization model

This paper introduces an optimization model developed for the steel industry, the most energy-intensive industrial sector in many countries. New methodologies for both aggregate production planning and energy supply management are developed to maximize profit while minimizing energy consumption and greenhouse gas (GHG) emissions. Moreover, uncertainties are considered, and robustly optimized strategies are identified for a grid-connected steel production company with an onsite energy generation system. The performance of the developed model is evaluated through extensive computational experiments. The results show how energy tariffs affect integrated production planning. Also, the selected production planning time depends on several factors such as electricity, labor, and maintenance costs, and pending orders that need to be fulfilled. The onsite power plant produces electricity when the electricity production cost is lower than the price of electricity provided by the connected grid. Furthermore, the company’s environmental policies influence the choice of energy generation technology. However, implementing an onsite energy generation is a viable policy that benefits both profitability and pollutant reduction. This study helps decision-making in production planning and energy supply management in the iron and steel industry (which can be applied to other industries too), ensuring environmental improvement at the lowest costs.

Approximate solution in robust multi-objective optimization and its application in portfolio optimization

Approximate solution in robust multi-objective optimization and its application in portfolio optimization

A multi-objective robust optimization to design a sustainable dairy supply chain under uncertain product demands, economic and social consideration. A real case study from Bulgaria

The operation of dairy supply chains (DSCs) is associated with the generation of significant amounts of wastewater pollutants and CO2 emissions. The presence of uncertainty regarding various parameters of the considered DSCs has an additional impact on their sustainability. The most efficient way to improve the sustainability of DSCs is by applying mathematical modeling approaches for optimal design of sustainable DSCs operating under uncertain conditions. The present study proposes an optimal model for design of a sustainable DSC for the production of two types of dairy products with a technology selection, which includes economic, environmental and social impact models, as well as dairy production model. The optimization model takes into account uncertainty in products demands, raw materials and products prices and social costs. The effectiveness of the proposed approach is proved in a real case of a DSC from Bulgaria, which includes suppliers of two types of milk and dairy plants for the production of two types of dairy products and markets. The obtained optimal values of the economic, environmental and social costs show that the application of the optimization approach results in sustainable solutions that do not significantly change with an increase in the uncertainty level of consideration of the uncertain parameters.

Intelligent pest trap monitoring under uncertainty in food industry

Pests bring significant losses in both quantity and quality of products in postharvest food facilities and their uncertain activities make imprecise obtained infestation information from trap monitoring, which causes unclear estimation on detriment costs, preventing effective pest management. Trap placement, which collects insect information through determining the exact location and number of traps, will enhance the precision for obtaining insect information. However, this method involves optimizing conflicting objectives. On one hand, smaller number of traps save the monitoring cost and enlarge the monitoring efficiency. On the other hand, number of traps may be insufficient and losses the monitoring accuracy. Existing studies to enhance the monitoring accuracy via both laboratory testing and high tech devices but they rarely considered economic efficacy in practice. In this paper, we characterize the trap placement problem as a Multi-Objective Optimization Problem under Uncertainty. Feedforward Neural Networks and Particle Swarm Optimization algorithms are used to design a tailored algorithm for modeling the conflicting objectives through process-based simulation. Then, robust multi-objective optimization evolutionary algorithm (RMOA) is developed to search for the Pareto optimal solution for the least number of traps settled at optimal positions. Results show the optimal and robust performance of exact trap locations to monitor insect infestation effectively under uncertain insect locations and dynamic insect growth. Compared to the studies that were mainly utilizing general algorisms for solving agricultural problems, the proposed model has implications for effective methods of capturing the characteristics of real-world insect monitoring applications in food industry.

Robust multi-objective building design optimization approach using NSGA-II algorithm considering weather and occupant uncertainties

A novel methodology is presented to evaluate how real-world operational uncertainties impact multi-objective building design optimization process. NSGA-II algorithm is applied across nine distinct operational scenarios, considering diverse weather datasets and energy usage patterns to optimize energy efficiency and thermal comfort in a residential building within India's hot and dry climate. The analysis considers 382,400 possible design options for each scenario. Using (10.5–12.5) % of model evaluations, NSGA-II significantly reduces Annual Cooling Energy Demand (ACED; 46.7%–58.7%) and Cooling Set-point Unmet Hours (CSUH; 70.92%–86.93%) to identify nine Pareto–Optimal solution sets. The final robust optimal design is confirmed by selecting the most repeated parameter settings, representing the highest robustness. Low thermal transmittance walls, double-glazed windows with a 10% window-to-wall ratio (WWR), deep overhangs, and a low solar heat gain coefficient are recognized as robust optimal building components. However, Air Conditioner (AC) sizing lacks robustness in minimizing energy demand across scenarios, requiring case-specific evaluation due to its sensitivity to weather and operational variations. Further, PAWN sensitivity analysis highlighted the substantial impact of WWR (KSWWR ∼ (0.46–0.66)) and wall type (KSWall ∼ (0.31–0.46)) on ACED and CSUH in chosen climate. AC operation is more important for predicting CSUH (KSAC ∼ 0.35) than ACED (KSAC ∼ 0.10).

Improvement Sets and Robust Multiobjective Optimization

Improvement Sets and Robust Multiobjective Optimization

Bi-level multi-objective robust optimization for performance improvements in integrated energy system with solar fuel production

The authors design an integrated energy system with solar fuel preparation as the core, integrating cooling, heating, electricity and fuel cell vehicles, aiming to meet the multiple energy needs of users. Fluctuations associated with renewable energy and energy demand pose a challenge to operational planning of systems. Unfortunately, common models do not take into account the impact of such fluctuations on system operation. To mitigate this challenge, the authors develop a bi-level multi-objective robust optimization model to obtain the optimal resilient dispatch of the system. The above model is transformed into a minimax optimization problem to alleviate the difficulties that solving it directly, and two nested algorithms are employed to maximize the energy conversion efficiency, minimize CO2 emissions and energy cost of the system. The energy, environmental and economic effects of system are compared using a robust optimization model and a deterministic optimization model. The operation performances of system in the bi-level robust optimization model are not all better than the results obtained from the deterministic model, because the robust optimization model considers the volatility of solar, wind, and customer loads during system operation. This research provides a feasible method to ensure the stability of the system.

Constrained multiobjective robust optimization of a bistable mechanism for inertial switch

Constrained multiobjective robust optimization of a bistable mechanism for inertial switch

Interval Pareto front-based multi-objective robust optimization for sensor placement in structural modal identification

Considering the multi-performance development of complex systems and the requirement of structural modal identification with typical uncertainties, the nominal single-objective optimization method is not suitable for sensor placement. Therefore, by combining conventional optimal sensor placement with non-probabilistic theory, this study proposes an uncertainty-oriented multi-objective robust optimization method for optimal sensor placement. The Fisher information matrix and ill-posedness comprise one eigenvalue-based optimization objective, and the mean and minimum off-diagonal values in the modal assurance criterion comprise another. Considering the high-cost limitation of the statistical method for handling uncertainties, uncertainty propagations are realized by a dimension-wise analysis with better accuracy and efficiency, thus avoiding the overestimation incurred by the classical Taylor expansion method. The multi-objective robust optimization is established by uncertain eigenvalue- and eigenvector-based indices with interval numbers and solved using the multi-objective optimization algorithm. Considering the solution sets located at the Pareto front, an interval possibility was developed using interval Pareto fronts to determine the optimal number of sensors. A numerical example demonstrated the validity of the proposed method with an optimal number of sensors and corresponding configurations.

Multi-objective Robust Optimization Over Time for Dynamic Disassembly Sequence Planning

Multi-objective Robust Optimization Over Time for Dynamic Disassembly Sequence Planning

A multi-objective robust evolutionary optimization approach applied to the multivariate helical milling process of super duplex steel

This work presents an approach for multivariate modeling and multi-objective robust evolutionary optimization for processes presenting as inputs control and noise variables and as outputs a set of correlated responses regarding process performance and product quality. Factor analysis is applied to obtain a transformed output set that is uncorrelated and retains most part of the information of the original outputs. Models that are robust regarding noise variables relating this transformed output set with the control variables are obtained through robust parameter design. Response models in the function of process and noise variables are obtained through the least squares method. Mean and variance models as a function of process variables are obtained through error propagation. Robust mean square error models are obtained to model the bias and variance of each latent variable. Finally, multi-objective evolutionary optimization is performed. Three evolutionary algorithms were applied and compared through hypervolume. The approach is applied in the helical milling of super duplex stainless steel UNS S32760. Tool overhang length, hole depth, and lubri-cooling flow rate were set as noise factors. Cutting forces were monitored, while roughness and roundness were measured to evaluate process performance and product quality. The three factors, obtained to represent the six outputs, guaranteed dimensionality reduction and minimized the redundancy in output space. The mean square error models obtained to these three factors as a function of control variables minimize both bias and variance regarding noise factors. The AGE-MOEA multi-objective evolutionary algorithm obtained the best performance considering the hypervolume. Some solutions with high trade-off were selected through pseudo-weights to aid the decision-making.

Multi-objective robust optimization of foam-filled double-hexagonal crash box using Taguchi-grey relational analysis

In this paper, a novel thin-walled double-hexagonal crash box is first proposed and then multi-objective robust optimized for better overall crashworthiness under multi-angle impact loading, using a proposed hybrid method combining aluminum foam-filling and Taguchi-grey relational analysis (GRA). Specifically, the finite element (FE) models of the regularly-shaped double-hexagonal column (DHC) extracted from original irregularly-shaped crash box under multi-angle impact loading, including hollow (H-DHC) and foam-filled (F-DHC), are first built and validated by experiments. On this basis, a comprehensive crashworthiness comparison is conducted to explore relative merits of F-DHC over original H-DHC under multi-angle impact loading. After that, the F-DHC is multi-objective robust optimized for maximizing overall specific energy absorption (SEAθ) and minimizing overall initial peak crushing force (IPCF0) simultaneously under multi-angle impact loading, using a hybrid method of Taguchi-GRA. At last, a bumper-crash box integrated crashworthiness analysis under multi-angle impact loading is executed to further verify the optimization. The optimal F-DHC and the optimized crash box within the optimal F-DHC demonstrate evident improvement of crashworthiness compared to their respective initial designs, indicating aluminum foam-filling combined with Taguchi-GRA could be an effective approach for multi-objective robust optimization of the novel crash box and other similar vehicle structures.

Position design of the trailing arm bushing on torsion beam suspension accounting for vehicle transient handling performance and uncertainties

The bushing of the trailing arm on torsion beam suspension plays a pivotal role in vehicle dynamic behavior. In this paper, the connection between bushing position and vehicle dynamic response is elucidated. According to the simulation results, the impact of the bushing position on the transient performance of vehicle is more pronounced at low handling frequencies, and different index under the same bushing position are not always optimal. To design the bushing position that is better for evaluation indexes, this paper formulates the design problem of the bushing position as a multi-objective optimization problem. Due to the influence of actual production and processing, inevitable errors in bushing position may result in vehicle performance not meeting design requirements. Therefore, this paper takes into consideration the uncertainties and conducts robust multi-objective optimization to design the bushing position. To address the computational burden associated with robust optimization, the RBF approximation model is developed in this paper. Finally, the optimization problem is solved with the NSGA-II intelligent algorithm. The optimization results show that the bushing position designed by robust multi-objective optimization results in vehicle with stronger anti-roll performance and better robustness for each evaluation index. It is more suitable for practical applications.

Multi-objective robust parameter optimization using the extended and weighted k-means (EWK-means) clustering in laser powder bed fusion (LPBF)

Metal additive manufacturing (AM) technology, especially laser powder bed fusion (LPBF), has received abundant interest from industries and the research community. Process optimization methods have thus multiplied to improve the overall quality of the final parts. However, little attention has been given to the quality repeatability issue. This paper proposes a novel multi-objective robust parameter optimization framework to explore optimal process parameters with respect to relative density and dimensional accuracy of LPBF-fabricated parts. Specifically, a modified k-means clustering, named the Extended and Weighted K-means (EWK-means), was constructed to simultaneously optimize the mean and the variance of the multiple responses. Experiments were conducted to verify the effectiveness of the proposed optimization framework. In addition, the effects of the process parameters, environment-related parameters, and physical properties on the hardness of the parts were analyzed using several machine learning models. The results showed that the proposed method achieved a set of optimal process parameters with better quality and satisfactory variability in the printed parts compared with other robust parameter optimization methods.

An integrated decision support framework for resilient vaccine supply chain network design

Designing resilient supply chain networks for vaccine development and distribution requires reliable and robust infrastructure. This stud develops a novel two-stage decision support framework for configuring multi-echelon Supply Chain Networks (SCNs), resilient supplier selection, and order allocation under uncertainty. Resilient supplier selection is done using a hybrid Multi-Criteria Decision-Making (MCDM) approach based on Best-Worst Method (BWM), Weighted Aggregated Sum Product Assessment (WASPAS), and Type-2 Neutrosophic Fuzzy Numbers (T2NN). A robust multi-objective optimization model is then built for order allocation considering resiliency scores, reliability of facilities, and uncertain supply and demand. The objectives are to minimize the total cost of SCN design, maximize the resiliency score, and maximize the reliability of SC, respectively. A Non-dominated Sorting Genetic Algorithm II (NSGA-II) is developed to tackle the problem on large scales, tuned by the Taguchi design technique. The NSGA-II solution is compared to the ε-constraint and Multi-objective Particle Swarm Optimization (MOPSO) solutions using test problems. We demonstrate the superiority of the suggested NSGA-II method over the two competing methods according to five performance metrics. A case study is then investigated to illustrate the applicability and effectiveness of the offered methodology for COVID-19 vaccine distribution in a developing country. It is revealed that the models and algorithms can treat the problem optimally, such that Germany is the main source (approximately 25.61%) while India does not contribute to the supply of vaccines.

Multi-objective topology optimization design of truss structures based on evidence theory under limited information

Uncertainty information is often limited and has great discreteness, how to ensure the structure having good robustness under epistemic uncertainty becomes a technical problem for multi-objective topology optimization design. In this paper, the evidence theory is used to quantify the epistemic uncertainty; the plausibility measurement expressing upper limit of failure probability is applied to be a new evaluation of reliability. The total weight and the reliability of structures are taken as the optimization objectives, and a multi-objective robust topology optimization design model based on evidence theory is proposed. Parallelization technique based differential evolution for multi-objective optimization (DEMO) is preferable to search above robust Pareto front due to its merits of non-requirement of any gradient and superior mechanisms of non-dominate strategy. In order to verify the effectiveness of the proposed method, the multi-objective reliability optimization of a wood truss structure is implemented with considering elastic modulus and structural load as uncertain variables. According to the optimization results, six same truss specimens are made for the static random loading test. The failure probability of the truss structure is judged by the stress and node displacement obtained from the test, so as to verify the feasibility of the reliability optimization method based on evidence theory. The experimental results also indicate that the proposed method can avoid the deviation of the optimization results caused by the fluctuation of epistemic uncertainty, and provide a new method for designers to make the optimization results robust even when the data information is insufficient and the cognitive level is limited.

Multiobjective robust optimization framework based on first and second order Taylor expansion applied to a vehicle suspension design

Multiobjective robust optimization framework based on first and second order Taylor expansion applied to a vehicle suspension design

Multi-objective distributionally robust approach for optimal location of renewable energy sources

Wind turbines and solar photovoltaic (PV) systems are intermittent and uncertain energy sources that disturb grid planning operations. In this paper, we establish a multi-objective distributionally robust optimization model for optimal locations of wind turbines and solar photovoltaics (PV) that minimize the variance of renewable energy sources and maximize power production. Moreover, this paper evaluates the accuracy of the Autoregressive Moving Average (ARMA), Deep Learning Gated Recurrent unit (GRU), and Deep Learning Long Short-Term Memory (LSTM) as forecasting models for wind speed and solar irradiation and compares their root mean square errors (RMSE). Using the forecasting error information, we characterize the uncertain variables in the ambiguity set, incorporating the bounds, means, and covariance values. Furthermore, we propose a modified multi-objective non-dominated sorting genetic algorithm (NSGA-II) approach to achieve a tractable Pareto front solution. To verify the effectiveness of the model, we use the actual candidate sites for wind turbines and solar photovoltaic (PV) systems in Saudi Arabia. The results demonstrate that our proposed model is an attractive and less conservative solution than a multi-objective robust optimization model when considering forecasting uncertainties.

Selecting Robust Multi-Objective Optimization of a MEMS Electro-Thermal Actuator Considering Multi-Source Uncertainties

The traditional robust multi-objective optimization (RMO) is computationally expensive, making it difficult to apply in engineering. To this end, a selecting RMO (SRMO) framework that considers multi-source uncertainties and the adaptive update of the surrogate model is proposed. The SRMO framework is more efficient and has been applied to optimize a micro-electro mechanical system (MEMS) electro-thermal actuator (ETA). The application of sensitivity analysis and surrogate modeling techniques improves efficiency in terms of reducing the problem dimensionality and the computational cost of fitness values, respectively. A multi-objective particle swarm optimization algorithm is adopted to find the better-performing points where time-consuming finite element analysis (FEA) will be performed, thus ensuring the prediction accuracy of the global surrogate (GS) model around them. A greedy algorithm is employed to select design points from the candidate solutions set to train local surrogate (LS) models that perform the robust evaluation taking into account multi-source uncertainties in dimensions, material parameters, and surface topography features. In the optimization of ETA, only 2.9% and 0.5% of the particles require performing FEA to update the GS model and train LS models, respectively. The final solutions show good robustness, and the effectiveness of the SRMO framework is verified by experiments and simulations.