Multi-objective Robust Optimization Research Articles

Robust multi-objective optimization for the Iranian electricity market considering green hydrogen and analyzing the performance of different demand response programs

Using renewable energy sources (RES) and green hydrogen has increased dramatically as one of the best solutions to global environmental issues. Applying demand response programs (DRPs) in this context could enhance the system’s efficiency. Evaluating different DRPs’ performances and assessing economic impacts on different parts of the electricity market is essential. The inherent uncertainty of RES and prices is inevitable in electricity markets. As a result of the lack of information, it is crucial to mitigate the risks as much as possible, such as risks related to changes in demand, unit outages, or other traders’ bid strategies. This research introduces a robust multi-objective optimization method to reach the most confident plan for the retailer based on uncertainty in RES and price. The integration of different DRPs is assessed according to the cost to retailers and benefits for consumers using a multi-objective model to survey the impacts of different parts’ decisions on each other. The trade-off among DRPs is considered in this model, and they are traded using a new model to illustrate the daily effect of these programs in monthly operations. This paper uses hydrogen storage (HS) integrated with PV as a distributed energy resource. As the Iranian electricity market has just been established, this research proposes a framework for decision-making in new electricity markets to join future smart energy systems. The mid-term pricing evaluates the system’s performance for more accurate monthly results. Also, the operation cost of the hydrogen storage is modeled to assess its performance in non-robust and robust scheduling. Mixed-integer linear programming (MILP) has been used to model this problem in GAMS. A developed linearizing method is considered with a controllable amount of errors to reduce the volume and time of the computation. Finally, the cost of consumers in non-robust and robust market planning in the presence of DRPs is reduced by 8.77 % and 9.66 %, respectively, and HS has a compelling performance in peak-shaving and load-shifting.

Experimental Investigation of Surface Roughness and Material Removal Rate in Wire EDM of Stainless Steel 304.

Its unexcelled mechanical and physical properties, in addition to its biocompatibility, have made stainless steel 304 a prime candidate for a wide range of applications. Among different manufacturing techniques, electrical discharge machining (EDM) has shown high potential in processing stainless steel 304 in a controllable manner. This paper reports the results of an experimental investigation into the effect of the process parameters on the obtainable surface roughness and material removal rate of stainless steel 304, when slotted using wire EDM. A full factorial design of the experiment was followed when conducting experimental trials in which the effects of the different levels of the five process parameters; applied voltage, traverse feed, pulse-on time, pulse-off time, and current intensity were investigated. The geometry of the cut slots was characterized using the MATLAB image processing toolbox to detect the edge and precise width of the cut slot along its entire length to determine the material removal rate. In addition, the surface roughness of the side walls of the slots were characterized, and the roughness average was evaluated for the range of the process parameters being examined. The effect of the five process parameters on both responses were studied, and the results revealed that the material removal rate is significantly influenced by feed (p-value = 9.72 × 10-29), followed by current tension (p-value = 6.02 × 10-7), and voltage (p-value = 3.77 × 10-5), while the most significant parameters affecting the surface roughness are current tension (p-value = 1.89 × 10-7), followed by pulse-on time (1.602 × 10-5), and pulse-off time (0.0204). The developed regression models and associated prediction plots offer a reliable tool to predict the effect of the process parameters, and thus enable the optimizing of their effects on both responses; surface roughness and material removal rate. The results also reveal the trade-off between the effect of significant process parameters on the material removal rate and surface roughness. This points out the need for a robust multi-objective optimization technique to identify the process window for obtaining high quality surfaces while keeping the material removal rate as high as possible.

A multi-objective robust grey wolf optimization model for the study of concrete mix using copper slag and glass powder

A multi-objective robust grey wolf optimization model for the study of concrete mix using copper slag and glass powder

Multi-objective optimal dispatching of virtual power plants considering source-load uncertainty in V2G mode

To solve the risks brought by the uncertainty of renewable energy output and load demand to the virtual power plant dispatch, a multi-objective information gap decision theory (IGDT) dispatching model for virtual power plants considering source-load uncertainty under vehicle-to-grid (V2G) is proposed. With the lowest system operating cost and carbon emission as the optimization objectives, the multi-objective robust optimization model for virtual power plants is constructed based on the uncertainties of wind output, photovoltaic output and load demand guided by the time of use price. The weights of uncertainties quantify the effects of uncertainty factors. The adaptive reference vector based constrained multi-objective evolutionary algorithm is used to solve it. The weight coefficients, evasion coefficients of uncertainties and the penetration rate of electric vehicles are analyzed for the optimal dispatching of the virtual power plant. The algorithm results show that the method can effectively achieve load-side peak shaving and valley filling and has superiority in terms of economy, environmental benefits, robustness and stability.

A sustainable scheduling system for medical equipment: Towards net zero goals for green healthcare.

Shortages of medical equipment, growth in medical waste and carbon emissions have increased healthcare pressures and has a huge impact on the environment. An efficient scheduling of medical equipment will effectively reduce the pressure on healthcare and improve the healthcare system's ability to respond to unexpected disasters. A medical equipment scheduling system was established to improve the sustainable utilization of medical equipment within the healthcare network and to reduce the carbon emissions of the healthcare process. First, this paper combines medical equipment information to establish a medical equipment scheduling decision model that considers pollution to filter qualified medical equipment for scheduling. Then, this paper constructs and solves a multi-objective robust optimization model by collecting the patient's travel information and the medical pressure information of each region. In addition, to meet dynamic healthcare needs, a dynamic medical equipment configuration framework was constructed to enhance the flexibility of equipment scheduling and the resilience of the healthcare network. Combined with case studies, the results show that the medical equipment scheduling system can help decision makers make quick scheduling decisions and achieve sustainable use of medical equipment, with a corresponding increase in medical equipment utilization of 12.25% and a reduction in carbon emissions of 26.50%. The study will help enhance healthcare resource utilization and contribute to the net-zero goal of green healthcare.

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.

Multi-objective robust optimization of profit for a naphtha cracking furnace considering uncertainties in the feed composition

A cracking furnace is the primary unit of a naphtha cracking center (NCC) to produce ethylene (EL) and propylene (PL); the yields of EL and PL depend on the naphtha composition of the NCC feedstock. However, the naphtha composition typically fluctuates depending on the feedstock suppliers, and the consequent uncertainties in the composition causes investment risks associated with the net profit. However, owing to the high dimensional uncertainties of the naphtha, conventional sampling (e.g. Monte Carlo method) based robust optimization is infeasible option due to high computational cost. In this study, we adopt the polynomial chaos expansion (PCE) to surrogate the system considering the uncertainty. Owing to the orthogonality of the PCE, the statistical moment of the PCE can be directly calculated without uncertainty sampling and iterative simulation. In this study, the PCE-based profit optimization model of the NCC furnace is developed as follows: Initially, we infer probability density functions (PDF) using industrial data to consider the uncertainty in the naphtha composition. Using the inferred PDF, we extract training dataset samples with coil outlet temperature (COT), product price, feed composition, and the net profit. Subsequently, using the training dataset, we develop a polynomial chaos expansion (PCE)-based surrogate model to predict the moments of the net profit, namely, the mean, variance, and skewness. Owing to the orthogonality of the model, the moments can be parameterized with only decision variables instead of computing the uncertainty. Finally, we incorporate the developed PCE-based model into a genetic algorithm to simultaneously optimize two conflicting objectives: maximizing the mean profit and minimizing the variance. The optimization results reveal the trade-off relationship between the mean profit and investment risk (variance and skewness) of the NCC process under feed uncertainty. Owing to the orthogonality, the optimal decision point can be provided with low computation time and high prediction accuracy compared with the sampling based optimization method. Considering the application of the proposed optimization model, we conduct case studies for two different scenarios of the product price. The optimal COTs for maximum mean profit with minimum variance of profit in the first and second scenarios range from 723 to 833 and 734 to 898 ℃, respectively. Therefore, the proposed model can quantitatively predict the mean profit with investment risk and help decision-makers select optimal operating conditions considering both the investment tendency and uncertainty in the naphtha composition.

A ROBUST OPTIMIZATION MODEL WITH TWO UNCERTAINTIES APPLIED TO SUPPLIER SELECTION

Under intense industry competition, decision makers must ensure that products demanded by consumers can be quickly produced with minimum production cost. However, because uncertainties are unavoidable and inevitably affect decision makers, numerous studies have discussed how to control uncertainties or minimize their effects. Multiple uncertainties that interact simultaneously may cause a combined effect in actual systems. Therefore, this study presents a robust optimization model with two uncertainties, extending the method of robust optimization with one uncertainty. To demonstrate the applicability of the proposed model with two uncertainties, this study uses the supplier selection problem with component purchase quantity allocation in supply chain management as an analysis case. This considers the reliability of production and transportation and develops a multi-objective robust optimization model with two uncertainties. In addition, a nondominated sorting genetic algorithm is proposed for solving the proposed multi-objective robust optimization model. The relationship between price of robustness and budget parameters is explored by considering the robust optimization model with production and transportation uncertainties proposed in this study. Finally, there is a comparative analysis between the results for price of robustness in the proposed two-uncertainty model and in the one-uncertainty model.

A robust multi-objective Bayesian optimization framework considering input uncertainty

Bayesian optimization is a popular tool for optimizing time-consuming objective functions with a limited number of function evaluations. In real-life applications like engineering design, the designer often wants to take multiple objectives as well as input uncertainty into account to find a set of robust solutions. While this is an active topic in single-objective Bayesian optimization, it is less investigated in the multi-objective case. We introduce a novel Bayesian optimization framework to perform multi-objective optimization considering input uncertainty. We propose a robust Gaussian Process model to infer the Bayes risk criterion to quantify robustness, and we develop a two-stage Bayesian optimization process to search for a robust Pareto frontier, i.e., solutions that have good average performance under input uncertainty. The complete framework supports various distributions of the input uncertainty and takes full advantage of parallel computing. We demonstrate the effectiveness of the framework through numerical benchmarks.

Multi-objective robust optimization using adaptive surrogate models for problems with mixed continuous-categorical parameters

Explicitly accounting for uncertainties is paramount to the safety of engineering structures. Optimization which is often carried out at the early stage of the structural design offers an ideal framework for this task. When the uncertainties are mainly affecting the objective function, robust design optimization is traditionally considered. This work further assumes the existence of multiple and competing objective functions that need to be dealt with simultaneously. The optimization problem is formulated by considering quantiles of the objective functions which allows for the combination of both optimality and robustness in a single metric. By introducing the concept of common random numbers, the resulting nested optimization problem may be solved using a general-purpose solver, herein the non-dominated sorting genetic algorithm (NSGA-II). The computational cost of such an approach is however a serious hurdle to its application in real-world problems. We therefore propose a surrogate-assisted approach using Kriging as an inexpensive approximation of the associated computational model. The proposed approach consists of sequentially carrying out NSGA-II while using an adaptively built Kriging model to estimate the quantiles. Finally, the methodology is adapted to account for mixed categorical-continuous parameters as the applications involve the selection of qualitative design parameters as well. The methodology is first applied to two analytical examples showing its efficiency. The third application relates to the selection of optimal renovation scenarios of a building considering both its life cycle cost and environmental impact. It shows that when it comes to renovation, the heating system replacement should be the priority.

6σ robust and optimal design of micro-motion platform based on kriging model

The two performance indexes of output displacement and inherent frequency are closely related to the design variables of the micro-motion platform. To address the problem that the design variables have certain fluctuations and the high nonlinearity between the design variables and the performance of the micro-motion platform, a multi-objective robust optimization method with high computational efficiency and high fitting accuracy is proposed. Based on the kriging agent model, the high-precision approximate functions of the key design variables with the output displacement and the inherent frequency are established by the Latin hypercube experimental design, respectively. With the maximum stress as the constraint, Monte Carlo simulation, 6σ robust design concept, and nondominated sorting genetic algorithm II (NSGA-II) multi-objective genetic algorithm are integrated to optimize the structure of the micro-motorized platform. The results show that compared with the traditional deterministic optimization method, the 6σ robust design considering parameter uncertainty effectively reduces the fluctuation of the performance of the micromotion platform and improves the resistance of the platform performance to disturbance under uncertainty.

Data-driven robust optimal control for nonlinear system with uncertain disturbances

Optimal control methods have attracted much attention for their promising performance in nonlinear systems. However, it is difficult to achieve satisfactory performance due to uncertain disturbances. To cope with this problem, a data-driven robust optimal control (DROC) method is proposed for uncertain nonlinear systems. The merits of the proposed DROC method are threefold: First, a data-driven evaluation strategy is introduced to capture the relationship between the approximating errors and the control variables. Then, the control performance indexes of nonlinear systems can be established within uncertain disturbances. Second, a multi-objective robust optimization algorithm is developed with a coevolution strategy. Then, robust optimal control laws can be obtained to improve the control performance. Third, the robust boundedness of DROC is discussed in theory. Then, the stability of the control systems can be guaranteed analytically. Finally, the effectiveness of DROC is illustrated with two multiple input multiple output second-order nonlinear systems. The optimal control performances are displayed in experiments to demonstrate the effectiveness of DROC.

Robust multi-objective optimization for islanded data center microgrid operations

Electricity cost has become a critical concern of data center operations with the rapid increasing of information processing demand. Data center microgrid (DCMG) is a promising way to reduce electric energy consumption from traditional fossil fuel generators and the billing cost, by effectively utilizing local renewable energy, e.g., wind power. However, uncertainties of wind power generation and real-time workload of data center would have significant impacts on the operational efficiency of DCMG, especially when it is in the island mode. For this reason, a novel affinely adjustable policy based robust multi-objective optimization model under flexible uncertainty set is proposed in this paper, which simultaneously optimizes wind power curtailment, the operation cost, and the over-plus level of computation resource, while considering uncertainties of both the wind power and real-time workload. Through numerical simulation studies, the validity of robust multi-objective optimization model for the island operation of DCMG is verified. Besides, the effectiveness of the proposed methods, i.e., the novel affinely adjustable policy and the flexible uncertainty set, in handling uncertainties are evaluated. Compared to the conventional robust multi-objective optimization model, the proposed approach reduces the operating costs of about 10% in average while maintaining similar reliability in numerical simulations. Moreover, the complex quantitative relationship among these multiple objectives is further investigated. Simulation results indicate the minimization of wind power curtailment and over-plus level of computation resource increases about 25% of the operation cost. These quantitative relationships can well support the decision making of DCMG operation management.

Nondeterministic multiobjective optimization of 3D printed ceramic tissue scaffolds

Despite significant advances in the design optimization of bone scaffolds for enhancing their biomechanical properties, the functionality of these synthetic constructs remains suboptimal. One of the main challenges in the structural optimization of bone scaffolds is associated with the large uncertainties caused by the manufacturing process, such as variations in scaffolds’ geometric features and constitutive material properties after fabrication. Unfortunately, such non-deterministic issues have not been considered in the existing optimization frameworks, thereby limiting their reliability. To address this challenge, a novel multiobjective robust optimization approach is proposed here such that the effects of uncertainties on the optimized design can be minimized. This study first conducted computational analyses of a parameterized ceramic scaffold model to determine its effective modulus, structural strength, and permeability. Then, surrogate models were constructed to formulate explicit mathematical relationships between the geometrical parameters (design variables) and mechanical and fluidic properties. The Non-Dominated Sorting Genetic Algorithm II (NSGA-II) was adopted to generate the robust Pareto solutions for an optimal set of trade-offs between the competing objective functions while ensuring the effects of the noise parameters to be minimal. Note that the nondeterministic optimization of tissue scaffold presented here is the first of its kind in open literature, which is expected to shed some light on this significant topic of scaffold design and additive manufacturing in a more realistic way.

Mixture Optimization of Cementitious Materials Using Machine Learning and Metaheuristic Algorithms: State of the Art and Future Prospects.

The hybrid optimization of modern cementitious materials requires concrete to meet many competing objectives (e.g., mechanical properties, cost, workability, environmental requirements, and durability). This paper reviews the current literature on optimizing mixing ratios using machine learning and metaheuristic optimization algorithms based on past studies on varying methods. In this review, we first discuss the conventional methods for mixing optimization of cementitious materials. Then, the problem expression of hybrid optimization is discussed, including decision variables, constraints, machine learning algorithms for modeling objectives, and metaheuristic optimization algorithms for searching the best mixture ratio. Finally, we explore the development prospects of this field, including, expanding the database by combining field data, considering more influencing variables, and considering more competitive targets in the production of functional cemented materials. In addition, to overcome the limitation of the swarm intelligence-based multi-objective optimization (MOO) algorithm in hybrid optimization, this paper proposes a new MOO algorithm based on individual intelligence (multi-objective beetle antenna search algorithm). The development of computationally efficient robust MOO models will continue to make progress in the field of hybrid optimization. This review is adapted for engineers and researchers who want to optimize the mixture proportions of cementitious materials using machine learning and metaheuristic algorithms.

Mathematical Modeling and Robust Multi-Objective Optimization of the Two-Dimensional Benzene Alkylation Reactor with Dry Gas

The benzene alkylation reactor using the dry gas is the most significant equipment in the ethylbenzene manufacturing process. In this paper, a two-dimensional homogeneous model is developed for steady state simulation of the industrial multi-stage catalytic reactor for ethylbenzene. The model validation on a practical benzene alkylation reactor shows the model is accurate and can calculate the hot spot temperatures. The composition of dry gas from upstream process varies with the operating conditions, which can cause unexpected hot spots in the reactor and catalyst deactivation. Considering the uncertainty in dry gas composition, a robust multi-objective optimization framework is proposed: first, the back-off in constraints is introduced to the multi-objective optimization problem to hedge against the worst case; then the optimal operating point can be selected using the multi-criteria decision-making. The reactor optimization objectives are maximizing selectivity of ethylene and conversion of ethylbenzene, and the distribution ratios of dry gas are defined as decision variables. Results of robust multi-objective optimization show the selectivity and conversion at the optimal operating point are 90.88% (decreased by 0.24% compared to the practical condition) and 99.94% (increased by 0.72%). Importantly, the proportion of violations of the hot spot constraints decreases from 13.7% of the traditional method to 3.8% by applying the proposed robust multi-objective optimization method.

Passenger Evacuation Path Planning in Subway Station Under Multiple Fires Based on Multiobjective Robust Optimization

The optimization of passenger evacuation path in subway station under emergencies directly affects the evacuation efficiency. Aiming at the evacuation scenario of passengers in subway station under multiple fires, a path planning method based on multi-objective robust optimization is established in this paper. The shortest total evacuation time, the minimum total risk and the minimum total congestion cost are taken as the objectives where the robustness of travel time and risk is also considered. NSGA-II algorithm is used to solve the model, the optimal Pareto solution under a certain robust control parameter is obtained according to the principle of minimizing the total cost function, and the overall optimization degree of route is obtained by using the evaluation index. Taking Qingdao May 4th Square subway station of China as a verification example, a fire diffusion model is built by using Pyrosim, and a passenger evacuation system is built by using Pathfinder. The multi-objective robust path optimization model of passenger evacuation is applied to the built passenger evacuation system under fires. The simulation results show that the overall optimization degree of route can reach 12.8%, when the time and risk robust control parameters are 30 and 30, respectively. The method can be used to guide the passenger evacuation in subway stations under multiple hazard sources and improve the safety index of passenger travel.

A data driven approach in less expensive robust transmitting coverage and power optimization

This paper aims the development of a new reduced-cost algorithm for a multi-objective robust transmitter placement under uncertainty. Toward this end, we propose a new hybrid Kriging/Grey Wolf Optimizer (GWO) approach combined with robust design optimization to estimate the set of Pareto frontier by searching robustness as well as accuracy (lower objective function) in a design space. We consider minimization of the energy power consumption for transmitting as well as maximization of signal coverage in a multi-objective robust optimization model. The reliability of the model to control signal overlap for multiple transmitting antennas is also provided. To smooth computational cost, the proposed method instead of evaluating all receiver test points in each optimization iteration approximates signal coverages using Kriging interpolation to obtain optimal transmitter positions. The results demonstrate the utility and the efficiency of the proposed method in rendering the robust optimal design and analyzing the sensitivity of the transmitter placement problem under practically less-expensive computational efforts (350% and 320% less than computational time elapsed using standalone GWO and NSGAII respectively).

Optimizing forest landscape composition for multiple ecosystem services based on uncertain stakeholder preferences

Determining the desirable composition of a forested landscape and its associated ecosystem services (ES) is challenging because the solutions must reconcile the preferences of various forest stakeholders and account for uncertain data. By combining multi-objective robust optimization with an online survey of forest professionals in Slovenia (n = 130) and forest professionals, forest scientists, nature conservationists and forest owners in Germany (n = 649) about optimal forest landscape composition, we derived compromise portfolios of forest types. These portfolios minimize the trade-offs between five ES (stopping avalanches, carbon storage, recreation, timber production and regulating flows of water), and account for the varying capacity of eight forest types to supply ES. The resulting optimized forest landscape compositions always comprised at least two forest types. In both countries, uneven-aged native deciduous and conifer mixed stands were prominent in the optimized portfolios. In Germany, however, the optimized portfolio also contained exotic species in mixtures, whereas forest stands without active management were notable for several ES in Slovenia. Unmanaged forest stands were also selected in the forest composition optimized for nature conservationists in Germany: the nature conservationists' portfolio diverged strongly from those of the other stakeholders. Our results illustrate that diversified forested landscapes provide multiple ES, but also secure the provision of a single ES when accounting for uncertainty. The optimal forest compositions obtained by multi-objective robust optimization are a starting point for participatory planning approaches to identify the most socially acceptable strategies for adapting forest management to an uncertain future.

Sufficient Optimality Conditions for a Robust Multiobjective Problem

In this note, we establish sufficient optimality conditions in terms of upper convexificators for a robust multiobjective optimization problem (UCP) involving nonsmooth/nonconvex real-valued functions. For this purpose, two kinds of generalized convex functions are defined in terms of upper convexificators.