Multiobjective Robustness Research Articles

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 robustness indicators for evaluation and exploration of design margins

ABSTRACT Specification of design margins serves the purpose of mitigating the ubiquitous uncertainty of the engineering design process. This paper addresses the twofold challenge of establishing a suitable concept of margins early in the design process, before the design is mature enough for detailed modelling, and evaluating margin inter-dependencies. As margins are often specified in an unstructured manner, there is a risk that one margin might lead to a need for more margins, e.g. tight tolerances, downstream. To avoid adverse margin dependencies, five Multi-Objective Robustness Indicators based on multi-objective design exploration are presented: The Relative Utopia Index, the Relative Pareto Shape Index, the Relative Compromise Index, the Relative Average Sensitivity Index, and the Average Feasible Variable Range. They aggregate optimality, trade-offs and sensitivity of design configurations. The indicators are applied to a case example, a flat head tank design, to illustrate the effect and inter-dependency of margin specification. Three margin types were identified. The first type affects only the Objective Space size, the second type affects only the Pareto set size and the third type affects Pareto set shape. The third margin type is the most critical as it leads to harder compromises between design objectives later in the design process.

Existence and connectedness of the l-minimal approximate solutions for set optimization problems: an application in generalized multiobjective robustness

Existence and connectedness of the l-minimal approximate solutions for set optimization problems: an application in generalized multiobjective robustness

Characterizations of multi-objective robustness solutions defined by Minkowski set difference

Characterizations of multi-objective robustness solutions defined by Minkowski set difference

Multi-Objective Optimization of the Robustness of Complex Networks Based on the Mixture of Weighted Surrogates

Network robustness is of paramount importance. Although great progress has been achieved in robustness optimization using single measures, such networks may still be vulnerable to many attack scenarios. Consequently, multi-objective network robustness optimization has recently garnered greater attention. A complex network structure plays an important role in both node-based and link-based attacks. In this paper, since multi-objective robustness optimization comes with a high computational cost, a surrogate model is adopted instead of network controllability robustness in the optimization process, and the Dempster–Shafer theory is used for selecting and mixing the surrogate models. The method has been validated on four types of synthetic networks, and the results show that the two selected surrogate models can effectively assist the multi-objective evolutionary algorithm in finding network structures with improved controllability robustness. The adaptive updating of surrogate models during the optimization process leads to better results than the selection of two surrogate models, albeit at the cost of longer processing times. Furthermore, the method demonstrated in this paper achieved better performance than existing methods, resulting in a marked increase in computational efficiency.

Robustness optimization for rapid prototyping of functional artifacts based on visualized computing digital twins

This study presents a robustness optimization method for rapid prototyping (RP) of functional artifacts based on visualized computing digital twins (VCDT). A generalized multiobjective robustness optimization model for RP of scheme design prototype was first built, where thermal, structural, and multidisciplinary knowledge could be integrated for visualization. To implement visualized computing, the membership function of fuzzy decision-making was optimized using a genetic algorithm. Transient thermodynamic, structural statics, and flow field analyses were conducted, especially for glass fiber composite materials, which have the characteristics of high strength, corrosion resistance, temperature resistance, dimensional stability, and electrical insulation. An electrothermal experiment was performed by measuring the temperature and changes in temperature during RP. Infrared thermographs were obtained using thermal field measurements to determine the temperature distribution. A numerical analysis of a lightweight ribbed ergonomic artifact is presented to illustrate the VCDT. Moreover, manufacturability was verified based on a thermal-solid coupled finite element analysis. The physical experiment and practice proved that the proposed VCDT provided a robust design paradigm for a layered RP between the steady balance of electrothermal regulation and manufacturing efficacy under hybrid uncertainties.

Assessment of an Optimal Design Method for a High-Energy Ultrasonic Igniter Based on Multi-Objective Robustness Optimization

The current deterministic optimization design method ignores uncertainties in the material properties and potential machining error which could lead to unreliable or unstable designs. To improve the design efficiency and anti-jamming ability of a high-energy ultrasonic igniter, a Six Sigma multi-objective robustness optimization design method based on the response surface model and the design of the experiment has been proposed. In this paper, the initial structural dimensions of a high-energy ultrasonic igniter have been obtained by employing one-dimensional longitudinal vibration theory. The finite element simulation method of COMSOL Multiphysics software has been verified by the finite element simulation results of ANSYS Workbench software. The optimal igniter design has been achieved by using the proposed method, which is based on the finite element method, the Optimal Latin Hypercube Design method, Grey Relational Analysis, the response surface model, the non-dominated sorting genetic algorithm, and the mean value method. Considering the influence of manufacturing errors on the igniter’s performance, the Six Sigma method was used to optimize the robustness of the igniter. The Eigenfrequency analysis and the vibration velocity ratio calculation were conducted to verify the design’s effectiveness. The results reveal that the longitudinal resonant frequency of the deterministic optimization scheme and the robustness optimization scheme are closer to the design’s target frequency. The relative error is less than 0.1%. Compared with the deterministic optimization scheme, the vibration velocity ratio of the robustness optimization scheme is 2.8, which is about 15.7% higher than that of the deterministic optimization scheme, and the quality level of the design targets is raised to above Six Sigma. The proposed method can provide an efficient and accurate optimal design for developing a new special piezoelectric transducer.

A multiobjective robustness methodology for a closed cycle supply chain design considering volume discount and economic benefits by reducing CO2 emissions of transportation modes

A multiobjective robustness methodology for a closed cycle supply chain design considering volume discount and economic benefits by reducing CO2 emissions of transportation modes

Design a Robust Logistics Network with an Artificial Physarum Swarm Algorithm

The robust optimization of logistics networks can improve the ability to provide sustainable service and business sustainability after uncertain disruptions. The existing works on the robust design of logistics networks insisted that it is very difficult to build a robust network topology, and this kind of optimization problem is an NP-hard problem that cannot be easily solved. In nature, Physarum often needs to build a robust and efficient topological network to complete the foraging process. Recently, some researchers used Physarum to build a robust transportation network in professional biological laboratories and received a good performance. Inspired by the foraging behavior of natural Physarum, we proposed a novel artificial Physarum swarm system to optimize the logistics network robustness just on a personal computer. In our study, first, the robustness optimization problem of a logistics network is described as a topology optimization model based on graph theory, and four robustness indicators are proposed to build a multi-objective robustness function of logistics network topology, including the relative robustness, the betweenness robustness, the edge robustness and the closeness robustness. Second, an artificial Physarum swarm system is developed to simulate the foraging behavior of a natural Physarum swarm to solve this kind of complex robust optimization problem. The proposed artificial Physarum swarm system can search for optimal solutions by expansion and contraction operations and the exchange of information with each other through a self-learning experience and neighbor-learning experiences. The plasmodium of Physarum forms the edges, and the external food sources simulate the logistics nodes. Third, an experimental example is designed on the basis of Mexico City to verify the proposed method, and the results reveal that the artificial Physarum swarm system can help us effectively improve the logistics network robustness under disruptions and receive a better performance than natural Physarum. The article may be helpful for both theory and practice to explore the robust optimization in logistics operation and provide engineers with an opportunity to resist logistics disruptions and risk loss by a novel artificial intelligence tool.

Robustness optimisation of opening casing wall-strengthened cylindrical pressure hulls based on 6σ

ABSTRACT This study determines a numerical analysis method for optimising the safety performance of a opening casing wall-strengthened cylindrical pressure hulls. After dimensional reduction of the design variables, the sample points are selected using the optimal Latin hypercube method, and fitted with the response surface model (RSM). After solving the deterministic lightweight optimisation problem, the weight of the hull is significantly reduced but the robustness is poor. Moreover, the robustness is little improved by a lightweight multi-response robustness optimisation on the pressure hull. Furthermore, the weight and ultimate strength are taken as the objective functions in a multi-response multi-objective robustness optimisation model. The optimisation obtains a Pareto solution set, from which four representative schemes are selected for further analysis. The σ levels of the constraints and ultimate strength increased to 8 without markedly changing the weight, indicating significant robustness improvement of the pressure hull.

Study on Structural Characteristics of Composite Smart Grille Based on Principal Component Analysis.

In recent years, many scholars have conducted in-depth and extensive research on the mechanical properties, preparation methods, and structural optimization of grid structural materials. In this paper, the structural characteristics of composite intelligent grid are studied by combining theoretical analysis with experiments. According to the existing conditions in the laboratory, the equilateral triangular grid structure experimental pieces were prepared. In this paper, principal component analysis combined with nearest neighbor method was used to detect the damage of composite plates. On this basis, the multiobjective robustness optimization of the structure is carried out based on artificial intelligence algorithm, which makes the structure quality and its sensitivity to uncertain parameters lower. Particle swarm optimization (PSO) is used in neural network training. The damage characteristics of different grid structures, different impact positions, and different impact energies were studied. The results show that the structural damage types, areas, and propagation characteristics are very different when the structure is impacted at different positions, which verifies that the grid structure has a good ability to limit the damage diffusion and shows that the grid structure has a good ability to resist damage.

Implementation of a novel Six Sigma multi-objective robustness optimization method based on the improved response surface model for bumper system design

In the vehicle industry, a car’s weight and crashworthiness are two conflicting indicators. Owing to uncertainties in loads, geometries, material properties, and operational conditions, deterministic optimization could lead to unreliable or unstable designs and increase the risk of design failure. This study combined the non-dominated sorting genetic algorithm (NSGA-II), Monte Carlo simulation with descriptive sampling (MCSDS), Six Sigma robustness approach, and improved response surface model (IRSM) to develop a general Six Sigma multi-objective reliability-based design optimization method algorithm (6σ-MORO). The proposed method was applied to examine the crashworthiness of a bumper system. In comparison to the response surface model (RSM), the accuracy of the IRSM was improved by only 10 groups of sample points, proving that the IRSM could construct each response surface function efficiently and accurately. Finally, based on the IRSM, a Six Sigma multi-objective robustness optimization was conducted to explore optimal reliability solutions. The results revealed that the proposed method improved the crashworthiness of the bumper system and also ensured its reliability. Note that the optimal design could be fairly promising for engineering design problems with high dimensions.

A Computationally Efficient Evolutionary Algorithm for Multiobjective Network Robustness Optimization

The robustness of complex networks is of great significance. Great achievements have been made in robustness optimization based on single measures, however, such networks may still be vulnerable to multiple attack scenarios. Therefore, recently, multiobjective robustness optimization of networks has received increasing attention. Nevertheless, several challenges remain to be addressed, including the different computational complexities in evaluating the objectives, insufficient diversity in the obtained networks, and high computational costs of the search process. In this article, we address the aforementioned challenges by developing a computationally efficient multiobjective optimization algorithm. Based on the unique features of complex networks, a new parallel fitness evaluation method guided by a network property parameter is designed and embedded in a reference vector-guided multiobjective evolutionary algorithm. In addition, a surrogate ensemble with heterogeneous inputs is constructed based on graph embedding information to efficiently estimate multiple robustness of networks. The proposed algorithm is validated on synthetic and real-world network data and our results show that the designed algorithm outperforms the state-of-the-art method with a marked improvement on the computational efficiency. Compared with other single-objective optimization methods, the proposed algorithm demonstrates a considerable exploitation ability.

The price of multiobjective robustness: Analyzing solution sets to uncertain multiobjective problems

Defining and finding robust efficient solutions to uncertain multiobjective optimization problems has been an issue of growing interest recently. Different concepts have been published defining what a “robust efficient” solution is. Each of these concepts leads to a different set of solutions, but it is difficult to visualize and understand the differences between these sets. In this paper we develop an approach for comparing such sets of robust efficient solutions, namely we analyze their outcomes under the nominal scenario and in the worst case using the upper set-less order from set-valued optimization. Analyzing the set of nominal efficient solutions, the set of minmax robust efficient solutions and different sets of lightly robust efficient solutions gives insight into robustness and nominal objective function values of these sets of solutions. Among others we can formally prove that lightly robust efficient solutions are good compromises between nominal efficient solutions and minmax robust efficient solutions. In addition, we also propose a measure to quantify the price of robustness of a single solution. Based on the measure, we propose two strategies which can be used to support a decision maker to find solutions to a multiobjective optimization problem under uncertainty. All our results are illustrated by examples.

Generalized multiobjective robustness and relations to set-valued optimization

In this paper, we provide the augmented weighted Tschebyscheff scalarization to computing robust efficient solutions for uncertain multiobjective optimization problems (UMOPs). Furthermore, we propose two generalized robustness concepts called the minmax certainly nondominated ordered robustness and the certainly less ordered robustness to general spaces and cones in the context of set orderings. Some explanations for both concepts are given, as well as the relations between them and several existing robustness are clearly described. Finally, the corresponding relationships to set-valued optimization are well revealed.

Characterizations of multiobjective robustness on vectorization counterparts

ABSTRACTIn this paper, we characterize robust solutions for uncertain multiobjective optimization problems on the basis of vectorization models by virtue of image space analysis. By introducing corrected image of original uncertain problem or the selected and corrected images of its robust counterpart, an equivalent relation between multiobjective robustness and the separation of two sets in the image space is well established. Moreover, by means of linear (vector/scalar) separation functions, some Lagrangian-type sufficient robust optimality conditions are presented. Especially, under suitable restriction assumptions, we derive Lagrangian-type necessary robust optimality conditions in terms of nonlinear separation functions. These results obtained in this paper extend and improve some existing ones recently. Finally, several examples are given to show the effectiveness of the conclusions.

Characterizations of Multiobjective Robustness via Oriented Distance Function and Image Space Analysis

In this paper, we characterize different kinds of multiobjective robustness concepts via the well-known oriented distance function. By using characterizations of several set relations via the oriented distance function, together with the help of image space analysis, we construct some suitable subsets of the scalarization image space to obtain equivalent characterizations for various robust solutions for uncertain multiobjective optimization problems based on a set approach.

Impact of mechanism vibration characteristics by joint clearance and optimization design of its multi-objective robustness

Joint clearances and friction characteristics significantly influence the mechanism vibration characteristics; for example: as for joint clearances, the shaft and bearing of its clearance joint collide to bring about the dynamic normal contact force and tangential coulomb friction force while the mechanism works; thus, the whole system may vibrate; moreover, the mechanism is under contact-impact with impact force constraint from free movement under action of the above dynamic forces; in addition, the mechanism topology structure also changes. The constraint relationship between joints may be established by a repeated complex nonlinear dynamic process (idle stroke - contact-impact - elastic compression - rebound – impact relief - idle stroke movement - contact-impact). Analysis of vibration characteristics of joint parts is still a challenging open task by far. The dynamic equations for any mechanism with clearance is often a set of strong coupling, high-dimensional and complex time-varying nonlinear differential equations which are solved very difficultly. Moreover, complicated chaotic motions very sensitive to initial values in impact and vibration due to clearance let high-precision simulation and prediction of their dynamic behaviors be more difficult; on the other hand, their subsequent wearing necessarily leads to some certain fluctuation of structure clearance parameters, which acts as one primary factor for vibration of the mechanical system. A dynamic model was established to the device for opening the deepwater robot cabin door with joint clearance by utilizing the finite element method and analysis was carried out to its vibration characteristics in this study. Moreover, its response model was carried out by utilizing the DOE method and then the robust optimization design was performed to sizes of the joint clearance and the friction coefficient change range so that the optimization design results may be regarded as reference data for selecting bearings and controlling manufacturing process parameters for the opening mechanism. Several optimization objectives such as x/y/z accelerations for various measuring points and dynamic reaction forces of mounting brackets, and a few constraints including manufacturing process were taken into account in the optimization models, which were solved by utilizing the multi-objective genetic algorithm (NSGA-II). The vibration characteristics of the optimized opening mechanism are superior to those of the original design. In addition, the numerical forecast results are in good agreement with the test results of the prototype.

A Unified Characterization of Multiobjective Robustness via Separation

This paper focuses on a unified approach to characterizing different kinds of multiobjective robustness concepts. Based on linear and nonlinear scalarization results for several set order relations, together with the help of image space analysis, some suitable subsets of scalarization image space are introduced to make equivalent characterizations for upper set (lower set, set, certainly, respectively) less ordered robustness for uncertain multiobjective optimization problems. In particular, the nonlinear scalarization functional plays a significant role in computing various multiobjective robust solutions. Finally, the corresponding examples are included to show the effectiveness of the results derived in this paper.

The relationship between multi-objective robustness concepts and set-valued optimization

Abstract In this paper, we discuss the connection between concepts of robustness for multi-objective optimization problems and set order relations. We extend some of the existing concepts to general spaces and cones using set relations. Furthermore, we derive new concepts of robustness for multi-objective optimization problems. We point out that robust multi-objective optimization can be interpreted as an application of set-valued optimization. Furthermore, we develop new algorithms for solving uncertain multi-objective optimization problems. These algorithms can be used in order to solve a special class of set-valued optimization problems.