Multi-objective Robust Optimization Research Articles

Review of Uncertainty, Carbon Emissions, Greenness Index, and Quality Issues in Green Supply Chains

The ability of closed-loop supply chains (CLSC) and reverse logistics (RL) to improve the triple bottom line (economic, social and environmental values) has increased the development of design and management models for CLSCs and RL. Consequently, there exists an extensive body of literature dedicated to exploring these supply and logistics issues. This paper reviews recent and relevant literature on CLSC and RL with an emphasis on uncertainty, carbon emissions, greenness index, return product quality and reliability considerations. The selected references are organized, reviewed, and analyzed to establish valuable mapping to highlight major findings. Finally, the outcomes are synthesized, and the primary research gaps are emphasized, pointing toward potential avenues for future investigation. These findings reveal that research efforts must be directed towards the development of multi-criteria greenness indices and multi-objective robust optimization models for uncertain quality and reliability of returns.

Multi-Objective Optimization of a Small-Scale ORC-VCC System Using Low-GWP Refrigerants

The increasing global demand for energy-efficient cooling systems, combined with the need to reduce greenhouse gas emissions, has led to growing interest in using low-GWP (global warming potential) refrigerants. This study conducts a multi-objective optimization of a small-scale organic Rankine cycle–vapor compression cycle (ORC-VCC) system, utilizing refrigerants R1233zd, R1244yd, and R1336mzz, both individually and in combination within ORC and VCC systems. The optimization was performed for nine distinct cases, with the goals of maximizing the coefficient of performance (COP), maximizing cooling power, and minimizing the pressure ratio in the compressor to enhance efficiency, cooling capacity, and mechanical reliability. The optimization employed the Non-dominated Sorting Genetic Algorithm III (NSGA-III), a robust multi-objective optimization technique that is well-suited for exploring complex, non-linear solution spaces. This approach effectively navigated trade-offs between competing objectives and identified optimal system configurations. Using this multi-objective approach, the system achieved a COP of 0.57, a pressure ratio around 3, and a cooling capacity exceeding 33 kW under the specified boundary conditions, leading to improved mechanical reliability, system simplicity, and longevity. Additionally, the system was optimized for operation with a cooling water temperature of 25 °C, reflecting realistic conditions for contemporary cooling applications.

Let decision-makers direct the search for robust solutions: An interactive framework for multiobjective robust optimization under deep uncertainty

The robust decision-making framework (RDM) has been extended to consider multiple objective functions and scenarios. However, the practical applications of these extensions are mostly limited to academic case studies. The main reasons are: (i) substantial cognitive load in tracking all the trade-offs across scenarios and the interplay between uncertainties and trade-offs, (ii) lack of decision-makers’ involvement in solution generation and confidence. To address these problems, this study proposes a novel interactive framework involving decision-makers in searching for the most preferred robust solutions utilizing interactive multiobjective optimization methods. The proposed interactive framework provides a learning phase for decision-makers to discover the problem characteristics, the feasibility of their preferences, and how uncertainty may affect the outcomes of a decision. This involvement and learning allow them to control and direct the multiobjective search during the solution generation process, boosting their confidence and assurance in implementing the identified robust solutions in practice.

A multi-objective robust dispatch strategy for renewable energy microgrids considering multiple uncertainties

The demand for low-carbon transformations and the uncertainty of renewable energy sources and loads present significant challenges for the optimal dispatch of microgrid. This study proposed a multi-objective robust dispatch strategy to reduce the risks associated with the uncertainty of renewable energy source output and loads while promoting low-carbon and economical microgrid operation. The economic emission dispatch problem for a microgrid was formulated as a multi-objective robust dual-layer optimization model. Consequently, a high-dimensional adjustable linear polyhedral uncertainty set was proposed to describe the uncertainty of renewable energy sources and loads. This study transformed the original model into an easy-to-solve single-layer second-order cone programming optimal power flow optimization model by employing second-order cone relaxation and duality transformation. Thereafter, a synthetic membership function was proposed to determine the optimal compromise solution. To determine the charging and discharging statuses of the battery storage system and the electricity traded between the microgrid and the external power grid, a battery storage system control strategy based on time-of-use electricity prices and real-time power flow calculations was proposed. Simulations conducted on a modified IEEE-30 bus system demonstrated that the proposed strategy effectively reduced the economic costs and carbon emissions of the microgrid by 8.23 % and 2.43 %, respectively.

Cathode-less RF plasma thruster design and optimisation for an atmosphere-breathing electric propulsion (ABEP) system

Atmosphere-breathing electric propulsion (ABEP) is a concept that ingests residual atmospheric gases as a source of propellant for an electric thruster, removing the need for onboard propellant storage. This would enable continuous low-thrust drag compensation, extending the lifetime of spacecraft in Very-Low Earth Orbit (VLEO); <250 km. VLEO is an appealing region for spacecraft operations, enabling new remote sensing missions with improved radiometric performance and spatial resolution, whilst reducing size, mass and power requirements, as well as mission cost. A preliminary design review and optimisation is therefore conducted for an ABEP system that uses the cathode-less radio frequency (RF) plasma thruster from Technology for Innovation & Propulsion (T4i) S.p.A. This removes the issue of thruster erosion by means of magnetic confinement and offers reduced susceptibility to varying atmospheric composition. A semi-empirical oxygen-nitrogen global source model (GSM) has been developed which considers the volume-averaged flux, momentum, and energy balance of the RF discharge. This includes a detailed chemistry model for the complex electron-molecular reactions and energy-loss channels of air plasma in the ionisation chamber. The GSM is coupled to an analytical model of flux balance for an air intake, verified by Direct Simulation Monte-Carlo (DSMC) simulation, to consider its design for maximum collection efficiency. This is then utilised in a robust multi-objective optimisation of the ABEP system, accounting also for spacecraft aerodynamics and power requirements.

Multi-objective robust optimization for optimal operation of gas-electricity networks with the penetration of integrated P2G and electric vehicle parking lots

Energy is considered one of the most important human needs. Therefore, various energy resources are exploited to meet human needs. Among all available resources, electricity and natural gas have accounted for the largest share in meeting these needs. Accordingly, management of these resources is one of the most important issues in different societies. This paper presents a multi-objective optimization model for optimal management and operation of the power distribution network and natural gas network as well as reducing carbon emissions. In this modeling, the natural gas network and distribution power network are connected through two electric vehicle parking lots (EVPLs) and two gas-fired units (GFUs), and these two EVPLs and GFUs exchange energy with each other. The EVPLs are equipped with various facilities including power-to-gas (P2G) systems and natural gas storage that facilitate energy exchange between the two networks. The modeling presented in this article is based on mixed integer linear programming, so the nonlinear equations including power flow equations, the Weymouth equation and the cost functions of GFUs are also linearized using various methods. It should be noted that the price of purchased electricity poses uncertainty which is modeled using robust optimization techniques. The proposed model is tested on the 33-bus power distribution network and the 20-node natural gas network. The results show that the presence of EVPLs significantly reduces the operational costs of the natural gas network and power distribution network and carbon emissions.

Solving multi-objective robust optimization problems via Stakelberg-based game model

Real-world multi-objective engineering problems frequently involve uncertainties stemming from environmental factors, production inaccuracies, and other sources. A critical aspect of addressing these problems, termed Multi-Objective Robust Optimization (MORO) problems, is the development of solutions that are both optimal and resilient to uncertainties. This paper proposes addressing these uncertainties through the application of Stackelberg game models, a novel approach involving the interaction of two players. The Leader searches for optimal and robust solutions and the Follower generates uncertainties based on the Leader’s chosen solutions. The Follower seeks to tackle the most challenging uncertainties associated with the Leader’s candidate solutions. Additionally, this paper introduces a novel metric to assess the robustness of a given set of solutions concerning specified uncertainties.Based on the proposed approach, a co-evolutionary algorithm is developed. A numerical study is then conducted to evaluate the algorithm by comparing its performance with those obtained by four benchmark algorithms on nine benchmark MORO problems. The numerical study also aims to assess its sensitivity to run parameter variations. The experimental results demonstrate the proposed approach’s effectiveness in identifying a non-dominated robust set of solutions.

Integrated management of electric vehicle sharing system operations and Internet of Vehicles energy scheduling

The sharing of electric vehicles and the Internet of Vehicles both positively impact societal benefits. However, the complexity, uncertainty and multi-directionality of transport–energy coupled systems greatly increase the difficulty of integrated management. This paper proposes a multi-objective robust optimization framework to solve the integrated management issue of electric vehicle sharing system operations and Internet of Vehicles energy scheduling under uncertain information, in which components such as renewable energy, microgrids, autonomous driving and their impacts are fully considered. To enhance the capability of integrated management and the synergy between supply and demand, a charging and discharging scheduling strategy that considers a converged demand response policy is proposed. Numerical analyses using real data provide valuable validation and insights. The proposed method and scheduling strategy enable the implementation of integrated management in photovoltaic microgrids and significantly improve economic and environmental benefits. Meanwhile, the microgrid system can operate in an ”off-grid mode”, which enables a multi-energy complementary mechanism. Additionally, electric vehicles release a great deal of scheduling flexibility through demand-side management, which combined with electric energy storage, can efficiently local consumption of photovoltaic power.

Multi-Objective Robust Design Optimization for Crashworthiness Enhancement of Hybrid 2D Triaxially Braided Composite Tube Using Evolutionary Algorithms.

An innovative optimal design framework is developed aiming at enhancing the crashworthiness while ensuring the lightweight design of a hybrid two-dimensional triaxial braided composite (2DTBC) tube, drawing insights from the mesostructure of the composite material. To achieve these goals, we first compile the essential mechanical properties of the 2DTBC using a concentric cylinder model (CCM) and an analytical laminate model. Subsequently, a kriging surrogate model to elucidate the intricate relationship between design variables and macroscopic crashworthiness is developed and validated. Finally, employing multi-objective evolutionary optimization, we identify Pareto optimal solutions, highlighting that reducing the total fiber volume and increasing the glass fiber content in the total fiber volume are crucial for optimal crashworthiness and the lightweight design of the hybrid 2DTBC tube. By integrating advanced predictive modeling techniques with multi-objective evolutionary optimization, the proposed approach not only sheds light on the fundamental principles governing the crashworthiness of hybrid 2DTBC but also provides valuable insights for the design of robust and lightweight composite structures.

Enhancing the Robustness of Rib-Groove Filling and Strain Homogeneity in the Isothermal Forging of Titanium Alloy Multi-Rib Components

Isothermal forging stands as an effective technology for the production of large-scale titanium alloy multi-rib components. However, challenges have persisted, including die underfilling and strain concentration due to the complex material flow and heterogeneous deformation within the forging die cavity. While approaches centered on optimized billet designs have mitigated these challenges, uncertainties in process parameters continue to introduce unacceptable variations in forming accuracy and stability. To tackle this issue, this study introduced a multi-objective robust optimization approach for billet design, accounting for the multi-rib eigenstructure and potential uncertainties. The approach includes finite element (FE) modeling for analyzing the die-filling and strain inhomogeneity within the multi-rib eigenstructure. Furthermore, it integrated image acquisition perception and feed back technologies (IAPF) for real-time monitoring of material flow and filling sequences within die rib-grooves, validating the accuracy of the FE modeling. By incorporating dimensional parameters of the billet and uncertainty factors, including friction, draft angle, forming temperature, speed, and deviations in billet and die, quantitative analyses on the rib-groove filling and strain inhomogeneity with fluctuation were conducted. Subsequently, a dual-response surface model was developed for statistical analysis of the cavity filling and strain homogeneity. Finally, the robust optimization was processed using a non-dominated sorting genetic algorithm II (NSGA-II) and validated using the IAPF technologies. The proposed approach enables robust design enhancements for rib-groove filling and strain homogeneity in titanium alloy multi-rib components.

Nature-Based Secondary Resource Recovery under Climate Change Uncertainty: A Robust Multi-Objective Optimisation Methodology

The management of high-volume (HV) waste poses a persistent challenge in sustainable materials management and represents an untapped opportunity in circular economy models. This study proposes a conceptual decision-making framework to operationalise a novel circular economy strategy for HV waste, involving temporary storage to facilitate nature-based secondary resource recovery. Using an illustrative case study of a candidate HV waste (legacy mining waste), we apply a robust multi-objective spatial optimisation approach at a national scale, employing an exact solution approach. Our methodology integrates mixed-integer linear programming to evaluate the economic viability, social benefits, and impacts of climate change uncertainties on nature-based solutions (NbS) implementation across diverse scenarios. The results demonstrate that NbS can enhance economic feasibility by incorporating carbon sequestration and employment benefits while demonstrating resilience against climate change projections to ensure long-term sustainability. The findings suggest that although NbS can improve the circular economy of HV nationally, it is essential to assess additional ecosystem services and address multiple uncertainties for effective macro-level sustainability assessment of HV management. This study offers a robust decision-making framework for policymakers and stakeholders to plan and implement nature-based circular economy strategies for HV waste streams at a national level while effectively managing long-term planning uncertainties.

Multi-objective robust optimization design framework for low-pollution emission burners

The optimal design of low-pollution emission burners plays an important role in controlling pollutant emissions of industrial equipment, and is crucial for the sustainable development of the national economy and environmental protection. However, many uncertain factors challenge the optimal design of low-pollution emission burners. The Latin hypercube sampling (LHS) method was used to obtain sampling data representing the distribution of the uncertain variable. The training dataset was obtained using the turbulent combustion coupling model. A high-precision sparse polynomial chaos expansion (PCE) model was constructed by the degree-adaptive scheme and least angle regression (LAR) algorithm. Furthermore, the Legendre polynomial is introduced to establish a continuous robust optimization model. The model is carried out by the non-dominated sorting genetic algorithm II (NSGA-II). The results show that the excess air coefficient of 1.227 is optimal. Compared with the excess air coefficient of 1.20 under the discrete robust optimization, the optimal coefficient can further reduce pollutant emissions and bring strong robustness to the ethylene cracking furnace. It has also been proven that the continuous robust optimization scheme improves the optimization granularity. Compared with discrete robust optimization, this method reduces the number of samples by 66.7 %.

Robust multi-objective optimization under multiple uncertainties using the CM-ROPAR approach: case study of water resources allocation in the Huaihe River basin

Abstract. Water resources managers need to make decisions in a constantly changing environment because the data relating to water resources are uncertain and imprecise. The Robust Optimization and Probabilistic Analysis of Robustness (ROPAR) algorithm is a well-suited tool for dealing with uncertainty. Still, the failure to consider multiple uncertainties and multi-objective robustness hinders the application of the ROPAR algorithm to practical problems. This paper proposes a robust optimization and robustness probabilistic analysis method that considers numerous uncertainties and multi-objective robustness for robust water resources allocation under uncertainty. The copula function is introduced for analyzing the probabilities of different scenarios. The robustness with respect to the two objective functions is analyzed separately, and the Pareto frontier of robustness is generated. The relationship between the robustness with respect to the two objective functions is used to evaluate water resources management strategies. Use of the method is illustrated in a case study of water resources allocation in the Huaihe River basin. The results demonstrate that the method opens a possibility for water managers to make more informed uncertainty-aware decisions.

Multi-objective robust dynamic pricing and operation strategy optimization for integrated energy system based on stackelberg game

The integrated energy system (IES) with hydrogen storage has become one of the most important developments in multi-energy coupling field, where the severe conflict of interests between different entities leads to great challenges to the economy and low-carbon operation. A multi-objective robust dynamic pricing and operation strategy optimization method based on the Stackelberg game is proposed for the hydrogen-containing energy storage (HES) IES. Firstly, the HES-IES trading framework is established based on the introduction of an integrated energy operator (IEO) and a load aggregator (LA). Secondly, a multi-objective robust Stackelberg game model is developed with the IEO as the leader and the LA as the follower, considering the minimization of operating costs and carbon emissions of the IEO and the minimization of integrated energy costs of the LA as the objectives. Finally, the compromise planning and the max-min fuzzy are addressed to solve the multi-objective model, which adopts the adaptive differential evolution (ADE) algorithm. In addition, the robust optimization (RO) with adjustable coefficients is employed to tackle uncertainties of source and load. The results show that this method can effectively balance the operating costs and carbon emissions of the system, improve the benefit of the IEO, reduce the costs of the LA, and avoid the uncertainty risk. Compared with traditional algorithms, the ADE algorithm has significant advantages in the number of iterations and solving time. In summary, the multi-objective robust dynamic pricing and operation strategy optimization proposed in this paper could effectively achieve the benefit balance between the IEO and the LA, also further improve the economy and robustness of the system and reduce carbon emissions.

Free vibration analysis and multi-objective robust optimization of three-dimensional pyramidal truss core sandwich plates with interval uncertain parameters

In this study, the free vibration analysis and multi-objective robust optimization of three-dimensional pyramidal truss core sandwich plates with interval uncertain parameters are fulfilled. The numerical model for free vibration of the plate is derived by combining the three-dimensional elasticity theory and Rayleigh-Ritz method, and the validity of the model is illustrated by numerical results. On this basis, considering various uncertainties within the plate, a new uncertainty-propagation analysis method is constructed by integrating the numerical model, interval-analysis model, kriging model and optimization. The one-dimensional and multi-source uncertainty-propagation analysis of the fundamental frequency is finished using this method, and the accuracy is proved by comparing with Monte Carlo simulation results. Meanwhile, to minimize the effects of uncertainties on the performance of plates at the design stage, an objective multi-objective robust optimization model with the objectives of maximizing the fundamental frequency and minimizing the robustness factor is established. Finally, an improved pelican optimization algorithm is proposed by introducing the improved opposition-based learning strategy, tracking strategy with step control and convergence strategy. And the Pareto front and corresponding design schemes applicable to different working environments are obtained without repeating the optimization.

Robust economic-resilient operational planning of distribution networks considering multi-microgrids black-start service

The increasing growth of natural disasters and cyber-physical attacks has made the development of resilience enhancement strategies for distribution networks a significant challenge for operators. Hence, in this paper, a black-start strategy based on multi-microgrids capability is developed as a two-layer preventive-corrective measure within a cost-effective energy management framework. The preventive layer includes regulating bilateral contracts between the active distribution network operator and the multi-microgrids owner pre-event. The corrective layer includes islanding microgrids during the event, calling the responsive loads, and economic-resilient operational planning of the active distribution network post-event. In this regard, a hybrid robust multi-objective optimization approach is applied to model the proposed black-start strategy as a risk-averse portfolio. The effectiveness of the developed model is tested on the modified IEEE 33-bus distribution network. Results such as an 83.38 % load recovery of the active distribution network, a 65.78 % reduction in distribution system operator costs, and a 131.28 % increase in multi-microgrid owner profit clearly illustrate the enhanced resilience of the active distribution network within an economically robust framework, which indicates the effectiveness of the proposed strategy.

Optimizing smart grid performance: A stochastic approach to renewable energy integration

Smart grids offer numerous possibilities for developing and executing sustainable and efficient electricity supply chains that exhibit increased resilience to disturbances. The exploration of the Electric Supply Chain Network (ESCND) design problem using smart grids has been limited. This article aims to tackle this challenge through the application of a robust multi-objective optimization approach, encompassing three economic goals (maximizing profit), environmental goals (minimizing greenhouse gas emissions), and resilience goals (maximizing network resilience). Additionally, the optimization process takes into account the dual objectives of maximizing efficiency and minimizing energy loss by incorporating relevant cost considerations. The proposed methodology incorporates distinctive features of smart grids, such as demandside management programs, microgrid structures, bidirectional distribution lines, and consumer–supplier nodes. It addresses various interconnected decisions, including location placement, capacity expansion, load allocation, and pricing. To solve the formulated model, a potent combination of multi-objective optimization methods based on cutting-plane algorithms and AUGMECON2 is introduced. Subsequently, the proposed model and solution approach are applied to a practical case study, and the obtained results are comprehensively examined. The findings suggest that, despite potential contradictions between economic and environmental objectives, the implementation of smart grids can concurrently improve environmental performance and network flexibility.

Multi-objective robust optimization of staff scheduling for emergency under stochastic demand

Large-scale centralized emergency rescue staff scheduling under emergencies has always been a challenge. With a specific focus on nucleic acid testing amid the COVID-19 pandemic, this study introduces a model that considers dynamic changes in staff supply at rescue points, the varying demand at sampling points over time, and their impact on the spread of the epidemic. We quantified the emergency weight of sampling points by the demand and the regional risk degree and adopted robust optimization with base constraint. With the objectives of minimum comprehensive service distance, maximum weighted value of demand satisfaction rate, and minimum loss caused by unmet demand, a multi-stage, multi-medical rescue point medical staff scheduling model for nucleic acid sampling points with the influence caused by uncertain demand disturbance was built, and an improved NSGA-II-HC algorithm was designed to solve the optimization problem under the influence of demand disturbances. The testing results proved that the NSGA-II-HC algorithm can tackle the issue of insufficient uniformity and diversity in the solution set of NSGA-II while Multi-Objective Particle Swarm Optimization (MOPSO) failed to do so. Taking the epidemic data of Guangzhou city in May 2021 as a case study, our model and algorithm are verified to be feasible. We offer a scheme selection strategy according to the loss degree of two conflicting objectives, the comprehensive service distance and shortage loss. The results suggest that, compared with the demand deterministic model, the decision mechanism under the robust model can reduce the deviation from the optimization objective due to the demand disruption.

Robust Multiobjective Optimization of Setback Feature in MEMS Safety and Arming Device Considering Parameters Uncertainty

Robust Multiobjective Optimization of Setback Feature in MEMS Safety and Arming Device Considering Parameters Uncertainty

A novel implicit decision variable classification approach for high-dimensional robust multi-objective optimization in order scheduling

This paper efficiently addresses the high-dimensional robust order scheduling problem. A novel algorithm named dynamic cooperative coevolution based on an implicit decision variable classification approach (DCC/IDVCA) is developed to search for robust order schedules. To significantly reduce the computational resources required for solving the high-dimensional robust order scheduling problem, we propose decomposing the original decision variables through implicit classification methods. First, a novel estimation method is introduced to evaluate the weighted contribution of variables to robustness. This method utilizes historical information, including the variation of the overall mean effective fitness and the frequency of variables being classified into highly robustness-related subcomponents in previous cycles, for evaluating their weighted contribution to robustness. Then, based on the corresponding weighted robustness contributions, the original variables are classified into highly and weakly robustness-related variables. Finally, these two types of variables are decomposed into highly and weakly robustness-related subgroups within a dynamic cooperative coevolution framework and optimized separately. In the experimental section, the proposed algorithm is applied to two practical order scheduling problems in discrete manufacturing industry. The experimental results demonstrate that the proposed algorithm achieves competitive outcomes compared to state-of-the-art high-dimensional robust multi-objective optimization algorithms.