Deterministic Multi-objective Optimization Research Articles

Steepest descent method for uncertain multiobjective optimization problems under finite uncertainty set

In this paper, a steepest descent method is developed for the robust counterpart of uncertain multiobjective optimization problems under finite uncertainty. The robust counterpart under consideration is the minimum of objective wise worst case, which is a nonsmooth deterministic multiobjective optimization problem. To solve this robust counterpart with the help of the steepest descent method, a subproblem is constructed and solved to find a descent direction. An Armijo-type inexact line search technique is employed to find a suitable step length. With the help of the descent direction and step length, a sequence is generated. The convergence of the proposed method is justified under some common assumptions. Finally, the algorithm is verified and compared with one existing method with the help of some numerical problems.

Robust retrofits for rural house envelope considering construction quality and occupant behavior uncertainties: A MOO-integrated Taguchi method

Due to the underdeveloped socio-economic environment, the uncertainties in construction quality and occupant behavior are more prominent for rural house than urban residential buildings. If these uncertainties are not considered, there are higher risks of inaccurate performance prediction, unrealistic retrofit decisions and causing serious performance gap. To address this challenge, this paper proposes a robust envelope retrofitting workflow that combine deterministic multi objective optimization (MOO) and Taguchi method. This workflow stands out for its utilizing mature deterministic MOO to fast approximate Alternative schemes, and utilizing Taguchi method to handle uncertainties, reducing some key disadvantages of traditional robust design optimization (RDO), such as heavy calculating burden, overly conservative solution and difficulty to predefine proper robust objectives or performance target value. The proposed workflow consists of four main steps, formulating Alternative schemes by deterministic MOO, screening highly sensitive uncertain parameters by uncertainty and global sensitivity analysis, formulating Typical uncertain scenarios by orthogonal experiments (OE) and calculating the comprehensive robustness indicators. Results show that by employing the proposed workflow, the selected robust optimal solution not only proves to be robust but also leads to preferable performance enhancements. Compared with the reference mode, the energy efficiency has been improved by 33.2% to 49.4%, the discomfort hours have been reduced by 0.4% to 26.5%, and the robustness has been enhanced by 36.1%.

Optimal dispatching of regional power grid considering vehicle network interaction.

When large-scale electric vehicles are connected to the grid for unordered charging, it will seriously affect the stability and security of the power system. To solve this problem, this paper proposes a regional power network optimization scheduling method considering vehicle network interaction. Initially, based on the user behavior characteristics and charging and discharging characteristics of electric vehicles, a charging and discharging behavior model of electric vehicles was established. Based on the Monte Carlo sampling algorithm, the scheduling upper and lower limits of each scheduling cycle of electric vehicles were described, and the scheduling potential of each scheduling cycle of electric vehicles was obtained. Then, the electricity price is then used as an incentive parameter to guide EV users to charge during periods of low electricity prices and participate in discharge during periods of peak electricity prices. Aiming at the highest economic efficiency, the best consumption effect of new energy and the smoothest demand-side power curve of regional power grid, a three-objective optimal dispatching model was established. In the later stage, uncertainty factors are taken into consideration by introducing the concept of interval numbers, and an interval multi-objective optimization dispatching model is established. The two dispatching models are solved by NSGA-II algorithm and improved NSGA-II algorithm, and the Pareto solution set is obtained. Finally, based on the analytic Hierarchy Process (AHP), the optimal scheduling scheme is determined. The Monte Carlo sampling method is used to simulate the user side charging demand, and the effectiveness of this method is verified. In addition, the results of the interval multi-objective optimization model and the deterministic multi-objective optimization model are compared, and it is proved that the solution results of the interval multi-objective model are more adaptive, practical and robust to the uncertain factors.

Decomposition-based interval multi-objective evolutionary algorithm with adaptive adjustment of weight vectors and neighborhoods

Interval multi-objective optimization problems (IMOPs) are one of the most critical optimization problems in practical applications. However, compared to deterministic multi-objective optimization problems (MOPs), there are few researchs addressing IMOP. In addition, the uncertainty contained in the problem makes the distribution of the population more challenging. Therefore, this paper proposed a decomposition-based interval multi-objective evolutionary algorithm with adaptive adjustment of weight vectors and neighborhoods (IMOEA/D-AWN). Firstly, an interval sparsity level function (ISL) is constructed to measure the density of individuals, and a comprehensive ranking of interval sparsity ranking and interval uncertainty ranking is proposed. For the purpose of improving the distribution of the population while reducing its uncertainty, based on the above comprehensive ranking, a new adaptive adjustment weight vector strategy guided by interval elite population is designed. Besides, an adaptive adjustment neighborhoods strategy is designed. This strategy adjusts individuals' neighborhood size according to the number of iterations to improve the efficiency of evolution. Finally, the IMOEA/D-AWN is evaluated on 17 interval benchmark test problems and a collaborative computation offloading optimization problem, and compared with four advanced multi-objective evolutionary algorithms with interval parameters (IMOEAs). Experimental results show that this algorithm performs well in convergence, diversity, and uncertainty.

Toward Performance Improvement of a Baculovirus–Insect Cell System under Uncertain Environment: A Robust Multiobjective Dynamic Optimization Approach for Semibatch Suspension Culture

The baculovirus expression vector system (BEVS) is one of the well-known versatile platforms for the recombinant protein/vaccine production. Mathematical modeling and optimization of a baculovirus–insect cell system can have significant industrial relevance as this reduces the number of expensive experiments and time involved in the experiment-based optimization. However, modeling and control of such a nonlinear system remains challenging due to the presence of uncertainties in the model. In this context, we propose a novel computational framework combining the principles of systems biology and dynamic optimization under uncertainty for optimizing a semibatch baculovirus–insect cell system. Toward this, first, a mathematical model replicating the dynamic experimental data on cell and virus growth was identified. Next, the proposed model was used for deterministic multiobjective dynamic optimization of the control variables, substrate, and multiplicity of infection (MOI) to achieve the conflicting objectives of productivity maximization and substrate minimization, simultaneously. Finally, based on the sensitivity analysis, six of the most influential parameters depicting model uncertainties have been considered for the robust multiobjective optimal control of the system. A comprehensive comparison displays up to 114% and 76% increases in the cell densities for the deterministic and stochastic semibatch processes, respectively, compared to the batch process. Semibatch operation also favors a minimum 40% reduction in MOI required to achieve the same level of infected cell density compared to the batch operation. This study provides a generic methodology for exhibiting a proof of concept that a semibatch suspension culture considering uncertainty in model parameters can give better productivity compared to a batch suspension culture for a BEVS.

A novel Bayesian approach for multi-objective stochastic simulation optimization

Multi-objective stochastic simulation optimization plays an important role in designing complex engineering systems. To identify optimal solutions via simulations, Bayesian optimization, which utilizes metamodels and an acquisition function to determine the next design point, has been popular in machine learning. However, studies on Bayesian optimization of multi-objective stochastic simulation are limited and can give undesired performances in terms of convergence and diversity metrics. Moreover, direct extension of Bayesian optimization algorithms of deterministic responses is unrealistic as stochastic responses are observed with heterogeneous noise. To handle these issues, we propose a novel framework for Bayesian optimization of multiple stochastic responses. Stochastic kriging metamodels are constructed independently for multiple stochastic responses considering heterogeneous simulation noise. To remove undesired effects of random simulation noise on comparing solutions, quantiles of stochastic kriging metamodels are employed to define the Pareto dominance. Acquisition functions are proposed by integrating the hypervolume-based and probability-of-improvement (PoI)-based acquisition functions from deterministic multi-objective optimization with typical acquisition functions from univariate stochastic response optimization. This not only enhances the convergence and diversity of Pareto-optimal solutions but also properly handles heterogeneous noise. Exact calculation formulas of the proposed algorithms are further developed. Typical numerical cases and two engineering examples (a rocket injector design example and a real inventory design scenario) demonstrate that the proposed algorithms give better Pareto-optimal solutions of which the proximity to true Pareto fronts and the diversity are improved.

Robust method for the identification of dynamical anisotropic flexible bearing parameters using multi-objective optimization and structural modification technique

In this paper, a deterministic multi-objective design optimization problem is addressed in order to identify the dynamical joint parameters such as mass, stiffness, damping and cross-coupling parameters. The proposed formulation uses directly the measured frequency transfer functions of the constrained system (with supports) and the mode of the unconstrained system (without supports) to predict the joint dynamic parameters. Since measurement noise often leads to erroneous identifications, the deterministic multi-objective optimization (DMOO) is used to eliminate the redistribution and amplification of noise due to the inversion of the frequency transfer function matrices. The proposed optimization design gives satisfactory results when using noise-free data as well as under realistic noise levels. A parametric study was also performed in order to assert the sensitivity of dynamical joints parameters against the level of noise. The results show also that the constrained non-dominated sorting genetic algorithm (C-NSGA-II) was capable to generate well-distributed reliable Pareto solutions.

Optimization Design for the Planetary Gear Train of an Electric Vehicle under Uncertainties

The planetary gear train is often used as the main device for decelerating and increasing the torque of the drive motor of electric vehicles. Considering the lightweight requirement and existing uncertainty in structural design, a multi-objective uncertainty optimization design (MUOD) framework is developed for the planetary gear train of the electric vehicle in this study. The volume and transmission efficiency of the planetary gear train are taken into consideration as optimization objectives. The manufacturing size, material, and load input of the planetary gear train are considered as uncertainties. An approximate direct decoupling model, based on subinterval Taylor expansion, is applied to evaluate the propagation of uncertainties. To improve the convergence ability of the multi-objective evolutionary algorithm, the improved non-dominated sorting genetic algorithm II (NSGA-II) is designed by using chaotic and adaptive strategies. The improved NSGA-II has better convergence efficiency than classical NSGA-II and multi-objective particle swarm optimization (MOPSO). In addition, the multi-criteria decision making (MCDM) method is applied to choose the most satisfactory solution in Pareto sets from the multi-objective evolutionary algorithm. Compared with the multi-objective deterministic optimization design (MDOD), the proposed MUOD framework has better reliability than MDOD under different uncertainty cases. This MUOD method enables further guidance pertaining to the uncertainty optimization design of transportation equipment, containing gear reduction mechanisms, in order to reduce the failure risk.

Integration of multi-objective reliability-based design optimization into thermal energy management: Application on phase change material-based heat sinks

Phase change material-based heat sinks are widely used in several industrial applications such as in the field of industrial automation and mechatronics. Nowadays, several researchers aim to obtain an optimal design of these systems in order to ameliorate its performances. Nevertheless, uncertainties are not considered for the majority of these studies. Furthermore, it has been confirmed that deterministic multi-objective optimization (DMOO) may lead to an unreliable design. In this study, a new methodology that leads to perform the Multi-Objective Reliability-Based Design Optimization (MORBDO) for thermal management of a passive cooling system is proposed. It consists in coupling the Finite Element Model (FEM), MORBDO procedures and surrogate approaches. Kriging predictor is used to construct metamodels and then validated using Cross-Validation (CV) and error predictions. A numerical investigation is carried out to study the different DMOO and MORBDO approaches. In this study, a 3D PCM-based round pin-fin heat sink is detailed. The aim of this problem is to minimize two objective functions: the cost (total volume V) and the final cooling time tf by considering both thermal and physical constraints. For this purpose, geometrical uncertain parameters are considered such as length L, height H and ϕ pin diameter of the heat sink. This study leads to develop a well-distributed reliable Pareto solutions by combining the Robust Hybrid Method (RHM) and the Constrained Non-dominated Sorting Genetic Algorithm (C-NSGA-II). Then, the efficiency of the proposed MORBDO-RHM approach for PCM-based heat sink is verified.

Reliability-based multi-objective optimization in tunneling alignment under uncertainty

This paper develops a framework of reliability-based multi-objective optimization (RBMO) in tunnel alignment. This study considers the two targets, the limit support pressure (LSP) and maximum ground surface deformation (MGSD), during the new tunnel’s excavation for safety and cost-saving purposes. The hybrid particle swarm optimization-neural network (PSO-NN) is used to construct the meta-model of the LSP and MGSD, based on the 100 groups of finite element numerical results of two tunnel’s excavation. The uncertainty from the soil material property and the meta-model has been considered in the RBMO as well. Through the Monte-Carlo simulation, the probability constraints in the RBMO are determined. Finally, this study entails an illustrative case to examine the superiority of the RBMO in comparison with the deterministic multi-objective optimization (DMO) and reliability-based single-objective optimization (RBSO). Through selecting the best solution of all the Pareto optimal solutions based on the Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS) approach, the optimized relative location of the newly built tunnel based on the RBMO is safer than that based on the RBSO under the tighter constraint for the LSP. In comparison with the RBSO, the RBMO generates a smaller LSP value with comparable MGSD value.

Robust Design and Optimization for a Permanent Magnet Vernier Machine With Hybrid Stator

This article tackles the design problem for a permanent magnet Vernier machine with hybrid stator to achieve a high structure robustness, enhanced torque and improved power factor simultaneously. The key of the solution is divided into two parts. One is to solve the conflict between torque and power factor by deterministic optimization. The other is to optimize its robustness with the help of robust optimization. Firstly, sensitivity analysis is adopted to stratify design parameters. Afterwards, a multi-objective deterministic optimization is applied to achieve an optimal balancing design between the two competing objectives, torque and power factor. However, the deterministic optimization ignores the size deviation in actual production, which most likely results in the poor structure robustness. Therefore, robustness analysis is employed to check its robustness. Then, through the robustness optimization, the robustness of this machine is further improved on the basis of the deterministic optimization. Finally, a prototype machine is built and the experiments on the prototype machine are carried out to verify the validity of the robust optimization.

A Deterministic Multi-Objective Optimization Process of Multi-Item Inventory Management in Fmcg Industry

The inventory management is the crucial part of any of the company’s, and nowadays, they find challenging to perceive out the optimal quantity level needed to overcome from shortages problem and also to reduce the wastages.In this paper, we used a multi-criteria decision-making for formulating the inventory management problem without shortages with the aim of optimizing profit and holding cost of the company simultaneously. We have also considered both ordering cost and storage cost as a non-linear function of the rapid stock level. In addition we used linear goal programming to find out the minimum optimal quantity level to put in the inventory to reduce out of stock situation. Finally, a numerical illustration of FMCG Company has been taken to show the application of proposed work.

A Robust Design Optimization Method Considering Correlated Intervals

In the robust design, correlations of uncertain parameters exist widely and have an influence on the results in most cases. It is essential to develop a robust design optimization method considering parametric correlation to future improve the analysis accuracy and engineering applicability. In this paper, a robust design optimization method based on multidimensional parallelepiped convex model is presented. Considering the effects of the interval uncertainties and their correlations, a robust design optimization model considering correlated intervals is established. In the established model, the average performance and robustness of the system response of concern are taken as the design optimization objectives, and the correlations among interval parameters are quantified by integrating the multidimensional parallelepiped convex model. And then, through an independence transforming procedure it can be converted into an independent interval model, which is ultimately converted into a deterministic multi-objective optimization model by using the interval possibility degree to cope with the uncertain constraints. Finally, the deterministic multi-objective optimization model is treated by coupling an efficient micro multi-objective genetic algorithm with the first order Taylor expansion. The feasibility and practicability of the proposed method are demonstrated by the numerical and engineering examples.

Rigorous modelling and deterministic multi-objective optimization of a super-critical CO2 power system based on equation of state and non-linear programming

CO2 power cycle is one of the promising sustainable technologies that generate power from fossil fuel, nuclear, solar, and various waste heat energy. The heat transfer process plays a key role in influencing the thermodynamic or thermo-economic performance of CO2 power cycle. The drastic variation of CO2 thermo-physical properties results in great difficulty in modeling and optimizing the heat exchangers and CO2 cycle configurations. In the present study, a mathematical programming method is proposed for the simulation and optimization of simple, regenerative, and recompression super-critical CO2 power systems. An accurate equation of state is applied to calculate CO2 properties, thereby enabling the performance indicator a continuous function of system parameters. A multi-objective non-linear programming model is formulated in GAMS for the automatic heat exchange pinch locating and system optimization. The simulation and optimization models are validated by comparison with REFPROP-based method. The maximum and average simulation errors are 0.064% and 0.015%. The thermal efficiency values achieved using the developed optimization method are improved by 6.14–10.85% compared with previous solutions. Then, a case study of a 573.15 K waste heat driven super-critical CO2 power system optimization is elaborated. Results show that pinch locating of all heat exchangers and system optimization can be simultaneously conducted within a short time. The solution optimality is validated by comparing the traditional pinch assignment concept and optimization method. Sensitivity analyses of heat source inlet temperature and heat sink temperature rise are also conducted. For both objectives of maximizing thermal efficiency and net power output, the recompression super-critical CO2 cycle features highest in thermal efficiency and the regenerative super-critical CO2 cycle features highest in net power output. Finally, a multi-objective optimization is conducted to achieve the Pareto Front for the studied three super-critical CO2 power systems.

Multi-Objective Robust Optimization for a Dual-Flux-Modulator Coaxial Magnetic Gear

The conventional multi-objective deterministic optimization (MODO) design may be less meaningful or even unacceptable when considering the perturbations of design parameters and reliability of the optimization results. In order to overcome the above-mentioned drawback and improve both the torque capability and permanent magnet (PM) utilization efficiency of dual-flux-modulator coaxial magnetic gear (DFM-CMG) simultaneously, a robust optimization design method is presented. In this method, the sigma criteria with the Monte Carlo simulation (MCS) are adopted to obtain the statistic samples addressing the effects of parametric uncertainties of DFM-CMG on the optimization results. Both 2-D and 3-D finite-element (FE) models of a DFM-CMG are first established, and the 3-D FE model is proven more accurate by the experiment and used for further optimization. Through the parametric study, five parameters are selected as the key design variables to establish the quadratic polynomial regression metamodels. Finally, the multi-objective particle swarm optimization algorithm with MCS is employed to conduct the multi-objective robust optimization (MORO) for DFM-CMG. Although the stall torque per PM consumption achieved by MORO with six sigma ( $6\sigma$ ) is little lower than that achieved by MODO, it still has a 21.4% growth than that of the initial design under the same constraint of the stall torque. Furthermore, the reliability and stability of the MORO results are much higher than those of the MODO results. The MORO method is significantly effective as the torque performance and robustness of DFM-CMG could be improved simultaneously.

Time-Dependent Reliability-Based Robust Design Optimization Using Evolutionary Algorithm

Due to the uncertain and dynamic parameters from design, manufacturing, and working conditions, many engineering structures usually show uncertain and dynamic properties. During the product design and development stages, designers often encounter reliability and robustness measures of dynamic uncertain structures. Time-varying and high nonlinear performance brings a new challenge for the reliability-based robust design optimization. This paper proposes a multi-objective integrated framework for time-dependent reliability-based robust design optimization and the corresponding algorithms. The integrated framework is first established by minimizing the mean value and coefficient of variation of the objective performance at the same time subject to time-dependent probabilistic constraints. The time-dependent probabilistic constraints are then converted into deterministic constraints using the dimension reduction method. The evolutionary multi-objective optimization algorithm is finally employed for the deterministic multi-objective optimization problem. Several examples are investigated to demonstrate the effectiveness of the proposed method.

An Effective Couple Method for Reliability-Based Multi-Objective Optimization of Truss Structures with Static and Dynamic Constraints

This paper proposes an effective couple method for solving reliability-based multi-objective optimization problems of truss structures with static and dynamic constraints. The proposed coupling method integrates a single-loop deterministic method (SLDM) into the nondominated sorting genetic algorithm II (NSGA-II) algorithm to give the so-called SLDM-NSGA-II. Thanks to the advantage of SLDM, the probabilistic constraints are treated as approximating deterministic constraints. And therefore the reliability-based multi-objective optimization problems can be transformed into the deterministic multi-objective optimization problems of which the computational cost is reduced significantly. In these reliability-based multi-objective optimization problems, the conflicting objective functions are to minimize the weight and the displacements of the truss. The design variables are cross-section areas of the bars and contraints include static and dynamic constraints. For reliability analysis, the effect of uncertainty of parameters such as force, added mass in the nodes, material properties and cross-section areas of the bars are taken into account. The effectiveness and reliability of the proposed method are demonstrated through three benchmark-type truss structures including a 10-bar planar truss, a 72-bar spatial truss and a 200-bar planar truss. Moreover, the influence of parameters on the reliability-based Pareto optimal fronts is also carried out.

A methodology for simulation-based, multiobjective gear design optimization

Design optimization of geared transmissions has become more of a necessity than ever before. Typically, conflicting design goals must be concurrently achieved. The difficulty of such a multiobjective design optimization problem is exacerbated by the fact that modern design practices rely on increasingly sophisticated, computationally-expensive simulation tools for tooth contact analysis. Their intrinsic nonlinearities add complexity to the problem, hampering gear designers’ efforts to obtain globally optimal solutions. Practical optimization problems of this class have often been solved by evolutionary algorithms, but their computational burden may well be inappropriate for CPU-intensive simulation models. The present work details an algorithmic framework inspired by deterministic multiobjective optimization methods, specially combined with a direct-search global optimization algorithm to obtain globally Pareto-optimal solutions. Nonlinear constraints are enforced through an exact penalty formulation. A comprehensive description of all theoretical and algorithmic details is provided, with the intention of enabling gear designers to implement or adapt the proposed methodology to their design optimization purposes. Two tests on a challenging gear design problem, namely ease-off topography optimization of a hypoid gear set for maximum efficiency and minimum contact stress, demonstrate that the proposed method can efficiently obtain solutions belonging to the global Pareto front.Declarations of interest: none.

Peat and pots: An application of robust multiobjective optimization to a mixing problem in agriculture

In horticultural settings, mixing problems arise when choosing a suitable growth substrate for potted plants. The aim of practitioners is to choose a pot and a growth substrate mixture such that environmental emissions and costs are simultaneously minimized. The impact of the plant’s quality on the selling price should also be considered. The decision problem outlined here is affected by two types of uncertainty. First, the problem parameters are uncertain due to the nature of the input data. For example variations in the agronomic characteristics of the growth media and in the weather conditions lead to imprecision. Second, the composition of the growth media is generally not implemented exactly due to variability in the mixing process itself. If these uncertainties are ignored, the problem can be addressed using deterministic multiobjective optimization. If these uncertainties are to be considered, the problem can be addressed by means of robust multiobjective optimization. In this paper, we apply both a deterministic and a robust approach to the problem, with and without uncertainty. We present a case study of an Italian plant nursery and we compare deterministic and robust solutions. Our results show that the robust solutions are preferable to the deterministic ones, and that the robust approach is indeed worth considering for horticultural mixing problems and might also be used effectively in other settings where similar mixing decisions arise.

An uncertain optimization method for overall ballistics based on stochastic programming and a neural network surrogate model

ABSTRACTA nonlinear stochastic programming method is proposed in this article to deal with the uncertain optimization problems of overall ballistics. First, a general overall ballistic dynamics model is achieved based on classical interior ballistics, projectile initial disturbance calculation model, exterior ballistics and firing dispersion calculation model. Secondly, the random characteristics of uncertainties are simulated using a hybrid probabilistic and interval model. Then, a nonlinear stochastic programming method is put forward by integrating a back-propagation neural network with the Monte Carlo method. Thus, the uncertain optimization problem is transformed into a deterministic multi-objective optimization problem by employing the mean value, the standard deviation, the probability and the expected loss function, and then the sorting and optimizing of design vectors are realized by the non-dominated sorting genetic algorithm-II. Finally, two numerical examples in practical engineering are presented to demonstrate the effectiveness and robustness of the proposed method.