Multi-objective Robust Optimization Research Articles

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.

Robust Multi-Objective Optimization Model for the Integration of Blood Production and Distribution Planning

Robust Multi-Objective Optimization Model for the Integration of Blood Production and Distribution Planning

Optimality Conditions of the Approximate Efficiency for Nonsmooth Robust Multiobjective Fractional Semi-Infinite Optimization Problems

This paper is devoted to the investigation of optimality conditions and saddle point theorems for robust approximate quasi-weak efficient solutions for a nonsmooth uncertain multiobjective fractional semi-infinite optimization problem (NUMFP). Firstly, a necessary optimality condition is established by using the properties of the Gerstewitz’s function. Furthermore, a kind of approximate pseudo/quasi-convex function is defined for the problem (NUMFP), and under its assumption, a sufficient optimality condition is obtained. Finally, we introduce the notion of a robust approximate quasi-weak saddle point to the problem (NUMFP) and prove corresponding saddle point theorems.

Multi-objective robust optimization strategy for community virtual cloud power plant considering different demand responses based on correlative confidence gap decision theory

To study the effect of different demand responses (DRs) on solving uncertainty, a multi-objective robust optimization model for a community virtual cloud power plant (CVCPP) is proposed based on the correlative confidence gap decision theory (CCGDT) and considering different DRs. Firstly, a CVCPP framework with multi-energy conversion equipment, multi-load DR, and a rapid energy scheduling information processing approach is proposed. Secondly, a two-layer community comprehensive demand response model is developed, which includes the CDR of CIES internal consumers and the CDR of the external energy network that connects to CIES. Thirdly, the CCGDT is presented using the confidence gap decision model and the time-varying Copula theory. A multi-objective robust economy-energy-environment optimization model is developed. Fourthly, the Pareto solution set is solved using the multi-objective grey wolf optimization algorithm. The full consistency method is employed to identify an optimal scheduling strategy for decision-makers. Finally, the model's validity is verified by a multi-scene case of a residential area. The results show that: 1) Whether uncertainty is considered or not, different DRs have varied effects on the results. Combination DR are better than single, and the same DR has different effects on different residents. 2) The multi-objective values improve after adopting the integrated energy system. The resident cost of various residents is decreased by 0.054%-12.07% when uncertainty is included. Meanwhile, the profit of energy providers is improved by 0.95%-10.98%, the carbon treatment amount is lowered by 26.85%-51.31%, and the renewable energy utilization rate is enhanced by 0.85%-7.94%. 3) The results with different significance have different robustness to uncertainty changes, and the greater the significance, the stronger the robustness. Therefore, the decision-maker can flexibly control the optimization result by setting the confidence of the target value. The model plays a good role in the economy-energy-environment aspects and provides a reference for community decision-makers to choose the comprehensive multi-objective energy scheduling strategy considering the uncertainty.

Robust Optimization Design of the Aerodynamic Shape and External Ballistics of a Pulse Trajectory Correction Projectile

To improve the tactical and technical performance of pulse correction projectiles while maintaining stability in uncertain conditions and considering practical engineering constraints, this study performs a multi-objective robust optimization design of the aerodynamic shape and external ballistics of a projectile. The study utilizes an aerodynamic force engineering algorithm and numerical trajectory calculations to obtain the projectile’s performance responses within the Latin hypercube design space. To enhance optimization efficiency, a stochastic Kriging surrogate model is established to capture the inherent uncertainty of limited input data. Ultimately, a Pareto optimal solution for the projectile is obtained using a non-dominated sorting multi-objective sparrow search algorithm. The results of this study demonstrate that the consideration of design uncertainty in the robust optimization of pulse correction projectiles leads to significant enhancements in both lateral correction ability and range while satisfying flight stability requirements. Moreover, when compared to deterministic optimization, the performance variability of the design is markedly improved. This research methodology provides valuable insights for optimizing the performance of pulse correction projectiles.

Multi-objective robust optimization for enhanced safety in large-diameter tunnel construction with interactive and explainable AI

Multi-objective robust optimization for enhanced safety in large-diameter tunnel construction with interactive and explainable AI

Multi-objective optimization ensembled generative design for large-diameter tunnel passing underneath existing tunnels

Automated and intelligent design in the construction industry has drawn considerable attention in these years along with the modern industrial revolutions. However, the achievement is strongly held up by the high unreliability in the underground space. To ameliorate this situation, this study develops a fuzzy robust multi-objective optimization (FRMOO) ensembled generative design system based on a large-diameter tunnel construction project with existing tunnels lying above. The robust part of the algorithm focuses on dealing with the unavoidable uncertainties from geotechnical conditions by seeking more conservative solutions. The fuzzy logic transforms the objective from minimizing the damage to seeking the most satisfactory solution by balancing all requirements from objectives and constraints. From evaluating the improvement in safety and the designer’s satisfaction degree simultaneously, the FRMOO approach reaches an overall improvement of 29.15%, which is found significantly superior to merely using robust optimization (18.93%) or simply using deterministic optimization (19.13%). Besides the FRMOO segment, the generative design system is also ensembled with a parametric modeling segment, which automatically builds a model based on the optimal solutions, and a user interface for the designer to interact with.

Improving robustness of water supply system using a multi-objective robust optimization framework

Due to the uncertainty of rainfall and water demand, the water supply of various participants has been challenged in such a way that this challenge has accelerated the failure of water supply system. Thus, this study proposes a multi-stage Adjustable Robust Optimization integrated to the multi-objective programming framework to drive water supply system to the failure safety zone and thereby improve robustness of system under different scenarios. Indeed, Adjustable Robust Optimization framework is applied to investigate the two uncertain factors of rainfall and water demand. A real arid area of Sistan basin in southeastern Iran is considered to analyze the proposed multi-objective programming model. Next, various comparative feasibilities under different levels of uncertainty are carried out to examine the robustness status in more detail. In the following, due to the deterioration of climatic patterns in the coming years, some managerial insights are highlighted. According to the final outputs, the domestic sector has reached more optimal value compared to that of the agricultural and industrial participants in all objectives due to less water intake, and as a result, it has a significant impact on the robustness of water supply system.

A robust optimization model for green supplier selection and order allocation in a closed-loop supply chain considering cap-and-trade mechanism

Due to increasing air pollution, which is a consequence of the environmental effects of production in various industries, green supply chain management (GSCM) has attracted the attention of both scholars and practitioners. Green supplier selection is an important problem in GSCM and seeks to satisfy a firm’s environmental goals as well as its economic targets. In this paper, for the first time, a green supplier selection problem considering both green and non-green evaluation criteria in a closed-loop supply chain is studied, and a cap-and-trade mechanism as a way of controlling the air pollution caused by manufacturers is proposed. To solve this particular problem, we propose a multi-objective robust optimization (RO) model. This specific model is an effective approach to handle uncertainty. A numerical example using randomly generated data, accompanied by subsequent discussion of the proposed approach, is deployed to validate the model. The results prove that the developed model for green supplier selection is able to effectively enhance the decision-making process of the experts. By illustrating the trade-off in robustness between the model and proposed solutions, as well as the effect of the deviation penalty on the closeness of results to the achieved solution, we show how firms can make optimal decisions when assigning the parameters. Furthermore, analyses show that decreasing the allowance amount (cap) and increasing the allowance prices in the cap-and-trade system escalate the firms’ costs, but lower the amount of carbon released. Finally, we show that the cap-and-trade mechanism results in a better solution in terms of the total utility of the supply chain compared to the penalty-based system for a specific range of carbon allowance.

Smart charging and discharging of electric vehicles based on multi-objective robust optimization in smart cities

The combination of low-carbon electricity and electric vehicles brings considerable economic and environmental benefits but also introduces challenges due to the complexity and uncertainty of system synergies in smart cities. To fully exploit the advantages of photovoltaic power generation and electric vehicles and to release the potential of electric vehicles as distributed energy storage facilities, this paper develops a multi-objective robust optimization framework that accounts for the benefits of multiple parties of smart charging and discharging systems and depicts a bounded uncertainty set based on partial statistical information from real data. The original model is scalarized and linearized using efficient methods such as max-ordering scalarization and the robust augmented weighted Tchebycheff to facilitate the solution. Moreover, a smart charging and discharging scheduling strategy based on a convergent demand response strategy is proposed to achieve better demand-side management. A case study, based on real data from Car2go and the solar radiation intensity in Portland, Oregon, shows that (1) The methodological framework effectively addresses the smart scheduling issue that considers the interests of multiple parties in complex systems with uncertainty, in contrast to conventional methods. (2) The designed convergent demand response strategy promotes photovoltaic power generation and charging demand synergy and achieves regulation effects such as peak shaving and valley filling. (3) The proposed method and strategy yield good results in multiple aspects, such as charging costs, load regulation, and clean energy utilization, while enhancing the economic and environmental benefits.

Multi-Objective Optimisation of a Novel Bypass Graft with a Spiral Ridge.

The low long-term patency of bypass grafts is a major concern for cardiovascular treatments. Unfavourable haemodynamic conditions in the proximity of distal anastomosis are closely related to thrombus creation and lumen lesions. Modern graft designs address this unfavourable haemodynamic environment with the introduction of a helical component in the flow field, either by means of out-of-plane helicity graft geometry or a spiral ridge. While the latter has been found to lack in performance when compared to the out-of-plane helicity designs, recent findings support the idea that the existing spiral ridge grafts can be further improved in performance through optimising relevant design parameters. In the current study, robust multi-objective optimisation techniques are implemented, covering a wide range of possible designs coupled with proven and well validated computational fluid dynamics (CFD) algorithms. It is shown that the final set of suggested design parameters could significantly improve haemodynamic performance and therefore could be used to enhance the design of spiral ridge bypass grafts.

Multi-variable approach to groundwater vulnerability elucidation: A risk-based multi-objective optimization model

Groundwater vulnerability mapping is essential in environmental management since there is an increase in contamination caused by excessive population growth. However, to our knowledge, there is rare research dedicated to optimizing the groundwater vulnerability models, considering risk conditions, using a robust multi-objective optimization algorithm coupled with a multi-criteria decision-making model (MCDM). This study filled this knowledge gap by developing an innovative hybrid risk-based multi-objective optimization model using three distinguished models. The first model generated two series of scenarios for rate modifications associated with two common contaminations, Nitrate and Sulfate, based on susceptibility index (SI) and DRASTICA models. The second model was a multi-objective optimization framework using non-dominated sorting genetic algorithms- II and III (NSGA-II and NSGA-III), considering uncertainties in the input rates by the conditional value-at-risk (CVaR) technique. Finally, the third model was a well-known MCDM model, the COmplex PRoportional ASsessment (COPRAS), which identified the best compromise solution among Pareto-optimal solutions for weights of the contaminations. Regarding the Sulfate's results, although the optimized DRASTICA model led to the same correlation as the initial model, 0.7, the optimized SI model increased the correlation to 0.8 compared to the initial model as 0.58. For the Nitrate, both the optimized SI and the optimized DRASTICA models raised the correlation to 0.6 and 0.7 compared to the initial model with a correlation value of 0.36, respectively. Hence, the best and the lowest correlation among the optimized models were between SI and Sulfate concentration and SI and Nitrate concentration, respectively.

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 multi-objective optimization of safety barriers performance parameters for NaTech scenarios risk assessment and management

Safety barriers are to be designed to bring the largest benefit in terms of accidental scenarios consequences mitigation at the most reasonable cost. In this paper, we formulate the problem of the identification of the optimal performance parameters of the barriers that can at the same time allow for the consequences mitigation of Natural Technological (NaTech) accidental scenarios at reasonable cost as a Multi-Objective Optimization (MOO) problem. The MOO is solved for a case study of literature, consisting in a chemical facility composed by three tanks filled with flammable substances and equipped with six safety barriers (active, passive and procedural), exposed to NaTech scenarios triggered by either severe floods or earthquakes. The performance of the barriers is evaluated by a phenomenological dynamic model that mimics the realistic response of the system. The uncertainty of the relevant parameters of the model (i.e., the response time of active and procedural barriers and the effectiveness of the barriers) is accounted for in the optimization, to provide robust solutions. Results for this case study suggest that the NaTech risk is optimally managed by improving the performances of four-out-of-six barriers (three active and one passive). Practical guidelines are provided to retrofit the safety barriers design.

The point-based robustness gap for uncertain multiobjective optimization

In robust single-objective optimization, the robustness gap is a measure of the distance between the robust optimal objective value and the optimal objective values of the scenarios. While robust multiobjective optimization is a growing field of study, no notion of a robustness gap has been proposed. A concept of a point-based robustness gap for uncertain multiobjective optimization problems is introduced. The gap is defined as the minimal distance between the robust Pareto set and the Pareto sets of the scenarios. It is shown that the gap is zero whenever the uncertainty is constraint-wise and objective-wise, supplementing a major result about the single-objective robustness gap. Because the distance between Pareto sets is hard to compute, lower and upper bounds on the gap are constructed for convex problems. Specific results about the zero gap and the bounds are presented for linear problems. Numerical examples are included.

Multiple criteria decision making and robust optimization to design a development plan for small and medium-sized enterprises in the east of Iran

This paper is based on a project of developing small and medium-sized enterprises in South Khorasan, Iran. A combination of multiple criteria decision making and robust multi-objective optimization is used for prioritizing industrial clusters and optimally assigning governmental funds. First, criteria for evaluating clusters are weighted using the best-worst method and clusters are prioritized using the VIKOR method. Second, governmental funding is assigned to the highest priority cluster using robust multi-objective mathematical programming. An innovative method is applied to identify the solution of the Pareto-front with the highest efficiency. In a case study from South Khorasan, computational results show that international market share attraction is the most important criterion and the age of the cluster is the least important criterion in the development of the industrial clusters. The cluster defined for barberry and jujube fruits is determined to be the first-ranked cluster. Most of the total available budget for the development of this cluster is assigned to action plans for marketing and trade development and for investment and financial planning. The proposed methodology thus successfully aided the decision makers to plan their work regarding funding of industrial clusters. We believe that the methodology can be applied as a general tool to help managers of industrial development to better assign governmental funding to develop industrial clusters.

Impacts of GCM credibility on hydropower production robustness under climate change: CMIP5 vs CMIP6

Climate change has emerged as a vital issue for hydropower management in the future. General circulation model (GCM)-informed climate is commonly applied to assess the impact of climate change on hydropower. However, the effect of GCM credibility on hydropower robustness has been rarely explored. One objective of this study is to seek valuable and robust policies, and the other is to evaluate the effects of GCM credibility on hydropower robustness based on the pre-optimized operating policies. A multi-objective robust optimization approach is first proposed to design operating policies by coupling stochastic simulation, heuristic parameterized policy search, and robust optimization. The GCM credibility is then evaluated based on multi-criteria ranking scores, and the hydropower robustness is illustrated by stepwise culling the lowest credible GCMs. The climate-induced impacts on hydropower robustness rely on an ensemble of 18 GCMs under three representative concentration pathways in Coupled Model Intercomparison Project 5 (CMIP5) and 17 GCMs under three shared socioeconomic pathways in CMIP6. The Longyangxia and Liujiaxia reservoirs in Upper Yellow River Basin are selected as a case study. The results show that (1) the proposed stochastic approach outperforms the deterministic approach regarding hydropower generation and firm output where the mean annual hydropower generation of stochastic operation is improved by -0.50- 19.53% compared to deterministic operation, indicating that the operating rules derived using historical stochastic inflows can adapt flexibly to future inflow changes than that using deterministic inflow. (2) CMIP6 maintains a more robust performance than CMIP5 with higher satisficing measures and lower regret measures. (3) Hydropower robustness tends to be higher when considering credibility and will be lower with time due to increasing uncertainty. Ignorance of GCM credibility may lead to biased climate change adaptation decisions.