Two-objective Optimization Problem Research Articles

Robust Performance-Based Design Optimization of Steel Moment Resisting Frames

The concept of performance-based design is considered in the framework of Robust Design Optimization (RDO) for the design of steel structures. The RDO problem is treated as a two-objective optimization problem where the initial construction cost and the variance of the maximum interstorey drift for the 10% in 50 years hazard level are considered as the problem objectives to be minimized. The structural performance is evaluated by means of the reliability demand and resistance methodology of the FEMA-350 guidelines in order to take into account both uncertainty and randomness in a consistent manner. The structure is designed to respond for different levels of seismic hazard levels with a desired confidence. The limit-state damage cost is used as a measure for the assessment of selected designs that have been obtained through the proposed RDO formulation. The non deterministic finite element problem encountered, is solved using the Monte Carlo Simulation method. The NSGA-II algorithm has been combined with the Evolution Strategies optimization algorithm for solving the two-objective optimization problem at hand.

Multi-Objective Optimization of Pressure Swing Adsorbers for Air Separation

Multi-objective optimization of the operation of a two-fixed-bed, four-stage pressure swing adsorber (PSA) unit for the separation of air is studied using the newly developed multi-objective optimization technique, the modified MOSA-aJG. The rigorous model developed for a single fixed-bed adsorption column is applicable to PSA units provided different and appropriate boundary and initial conditions are used for the four stages, namely, pressurization, adsorption, blowdown, and purge. Seven of the nine model parameters are the same as obtained earlier for a single-bed O2−N2−Zeolite 5A unit. The remaining two parameters are tuned using one set of experimental data on a PSA unit with the same system. Reasonably good agreement is observed between simulation and experimental results. Two two-objective optimization problems and a four-objective problem are formulated for the operation of a PSA. These are solved to obtain optimal values of the adsorption pressure, desorption pressure, adsorption and purge times, and input flow rates during the adsorption and purge stages. The effects of different decision variables on the several objective functions are discussed in detail.

Optimal robust linearized impulsive rendezvous

The optimization of time-fixed linearized impulsive rendezvous with control uncertainty is investigated. One performance index related to the variances of the terminal state error is defined as the performance index of robustness which is calculated by linear covariance method. The two-objective optimization problem of minimizing the total characteristic velocity and the performance index of robustness is formulated based on the Clohessy–Wiltshire (C-W) system and solved by the nondominated sorting genetic algorithm. The Pareto-optimal solution sets of one homing rendezvous mission are provided and the Pareto optimality is verified by comparing with the fuel-optimal and the robustness-optimal solutions. It is shown that the proposed approach can quickly investigate the relation between the fuel cost and the trajectory robustness, besides evaluate different rendezvous maneuver schemes.

226 一般化伝達剛性係数法を利用した骨組構造物の位相最適化

This paper presents a topology optimization method of framework structures using the genetic algorithm and the generalized transfer stiffness coefficient method (GTSCM). The two-objective optimization problem that minimizes the total weight and maximizes the first natural frequency under the displacement constraint and the stress constraint is formulated, and the genetic algorithm based on the ground structure approach is developed. In the present algorithm, the merits of the GTSCM are utilized in order to improve the computation efficiently, and the effective methods of genetic operations for the topology optimization problem are proposed.

A Design Support Method for Component Arrangement of Space Satellites Using Bi-level Optimization (Preliminary Study on a Piggyback Satellite Design Case)

Component arrangement in space satellites is an important design process at the conceptual design stage, since component location influences a range of fundamental characteristics of a satellite, and design change in the latter detailed design stage is quite difficult. In this research, a preliminary study for development of systematic design methods for such component arrangement based on optimization techniques is conducted. The component arrangement design problem is formulated as a two-objective optimization problem, and this optimization problem is solved using a bi-level optimization procedure. At the first optimization step, the genetic algorithm is applied to obtain the approximate location of components, and the second step uses the successive quadratic programming to determine the detailed positions of the components. Furthermore, this paper proposes a competent genotype representation technique for handling a variety of component arrangement patterns during the genetic algorithm optimization process. Finally, the proposed method is applied to component arrangement design of a practical small piggyback satellite in order to illustrate effectiveness of the proposed method.

Robust seismic design optimization of steel structures

Stochastic performance measures can be taken into account, in structural optimization, using two distinct formulations: robust design optimization (RDO) and reliability-based design optimization (RBDO). According to a RDO formulation, it is desired to obtain solutions insensitive to the uncontrollable parameter variation. In the present study, the solution of a structural robust design problem formulated as a two-objective optimization problem is addressed, where cross-sectional dimensions, material properties and earthquake loading are considered as random variables. Additionally, a two-objective deterministic-based optimization (DBO) problem is also considered. In particular, the DBO and RDO formulations are employed for assessing the Greek national seismic design code for steel structural buildings with respect to the behavioral factor considered. The limit-state-dependent cost is used as a measure of assessment. The stochastic finite element problem is solved using the Monte Carlo Simulation method, while a modified NSGA-II algorithm is employed for solving the two-objective optimization problem.

The optimization of the variable binder force in U-shaped forming with uncertain friction coefficient

In this paper, an uncertain optimization method is suggested to obtain the optimal variable binder force in U-shaped forming. The friction coefficient is regarded as the uncertain coefficient, and the stepped variable binder force model is used. The finite element method is employed to simulate the forming process, and an uncertain objective function which represents the springback magnitude is created. The uncertain friction coefficient is treated as an interval, no need to know its probabilistic distribution. Through a nonlinear interval number programming method, the uncertain optimization problem is converted into a deterministic two-objective optimization problem. A hybrid optimization algorithm based on the intergeneration projection genetic algorithm and neural network is used to obtain the optimum. The presented method is applied to optimize the variable binder force parameters of the model from NUMISHEET’93. The forming quality based on the optimal variable binder force from the presented method is compared with constant binder force. The results indicate that the presented method can find the fair variable binder force to obtain both of the small springback and strain under the uncertain friction coefficient.

Simultaneous optimization of the performance of flotation circuits and their simplification using the jumping gene adaptations of genetic algorithm-II: More complex problems

The binary-coded elitist non-dominated sorting genetic algorithm with the modified jumping gene operator (NSGA-II-mJG) is used to obtain global optimal solutions of flotation circuits. Several single-objective and multi-objective optimization problems are solved using the interconnecting cell linkage parameters (fraction flow rates) and the mean cell residence times as the decision variables. In the single-objective problem, the overall recovery of the concentrate stream is maximized for a desired grade of the concentrate. Two two-objective optimization problems are then solved. In one, the number of non-linking streams and the overall recovery of the concentrate are maximized simultaneously. This gives several simple circuits in a systematic manner with only marginally lower recoveries. In the other two-objective optimization problem, the overall recovery of the concentrate is maximized while the total cell volume is minimized. A three-objective problem (maximization of the overall recovery of the concentrate, maximization of the number of non-linking streams and minimization of the total cell volume) is then solved. All the problems constrain the grade of the product to lie at a fixed value. Finally, a complex and computationally intensive four-objective optimization problem is solved. The solution of several practical optimization problems in this study helps develop useful insights into the optimal solutions.

The influence of the fitness evaluation method on the performance of multiobjective search algorithms

In this paper we are concerned with finding the Pareto optimal front or a good approximation to it. Since non-dominated solutions represent the goal in multiobjective optimisation, the dominance relation is frequently used to establish preference between solutions during the search. Recently, relaxed forms of the dominance relation have been proposed in the literature for improving the performance of multiobjective search methods. This paper investigates the influence of different fitness evaluation methods on the performance of two multiobjective methodologies when applied to a highly constrained two-objective optimisation problem. The two algorithms are: the Pareto archive evolutionary strategy and a population-based annealing algorithm. We demonstrate here, on a highly constrained problem, that the method used to evaluate the fitness of candidate solutions during the search affects the performance of both algorithms and it appears that the dominance relation is not always the best method to use.

05/02545 Electricity estimation using genetic algorithm approach: a case study of Turkey

The elitist non-dominated sorting genetic algorithm with the modified jumping gene operator (NSGA-II-mJG) is used to optimize the performance of froth flotation circuits. Four example optimization problems (Mehrotra and Kapur, 1974; Green, 1984; Dey et al., 1989) [Mehrotra, S.P., Kapur, P.C., 1974. Optimal–sub-optimal synthesis and design of flotation circuits. Sep. Sci. 9, 167–184; Green, J.C.A., 1984. The optimization of flotation networks. Int. J. Miner. Process. 13, 83–103; Dey, A.K., Kapur, P.C., Mehrotra, S.P., 1989. A search strategy for optimization of flotation circuits. Int. J. Miner. Process. 26, 73-93.] of varying complexity are solved using single-objective functions. In one example, the overall recovery of the concentrate stream is maximized for a desired grade of the concentrate and a fixed total cell volume. The interconnecting cell linkage parameters (fraction flow rates) and the mean cell residence times are the decision variables. In all these cases, the optimal solutions obtained using NSGA-II-mJG are superior to those obtained by earlier techniques (which converged to local optimal solutions). Thereafter, a few two-objective optimization problems are solved. In these, the performance of the circuit is optimized, and simultaneously, the number of connecting streams is minimized so as to give simple circuits. Pareto optimal sets of equally good (non-dominating) solutions are obtained. This is probably the first study involving the multi-objective optimization of flotation circuits with one aim being to simplify them.

Simultaneous optimization of the performance of flotation circuits and their simplification using the jumping gene adaptations of genetic algorithm

The elitist non-dominated sorting genetic algorithm with the modified jumping gene operator (NSGA-II-mJG) is used to optimize the performance of froth flotation circuits. Four example optimization problems (Mehrotra and Kapur, 1974; Green, 1984; Dey et al., 1989) [Mehrotra, S.P., Kapur, P.C., 1974. Optimal–sub-optimal synthesis and design of flotation circuits. Sep. Sci. 9, 167–184; Green, J.C.A., 1984. The optimization of flotation networks. Int. J. Miner. Process. 13, 83–103; Dey, A.K., Kapur, P.C., Mehrotra, S.P., 1989. A search strategy for optimization of flotation circuits. Int. J. Miner. Process. 26, 73-93.] of varying complexity are solved using single-objective functions. In one example, the overall recovery of the concentrate stream is maximized for a desired grade of the concentrate and a fixed total cell volume. The interconnecting cell linkage parameters (fraction flow rates) and the mean cell residence times are the decision variables. In all these cases, the optimal solutions obtained using NSGA-II-mJG are superior to those obtained by earlier techniques (which converged to local optimal solutions). Thereafter, a few two-objective optimization problems are solved. In these, the performance of the circuit is optimized, and simultaneously, the number of connecting streams is minimized so as to give simple circuits. Pareto optimal sets of equally good (non-dominating) solutions are obtained. This is probably the first study involving the multi-objective optimization of flotation circuits with one aim being to simplify them.

Optimal Operation of an Industrial-Scale Parex Process for the Recovery of p-Xylene from a Mixture of C8 Aromatics

Multi-objective optimization for the operation and design of an industrial-scale Parex process for the recovery of p-xylene from a mixture of C8 aromatics has been studied using a state-of-the-art optimization technique: the Elitist Nondominated Sorting Genetic Algorithm with Jumping Genes (NSGA-II-JG, binary coded). An industrial-scale Parex process, which is based on simulated moving bed (SMB) technology, was simulated using an available mathematical model. The simulation results were verified with the reported industrial data. Subsequently, few two-objective optimization problems were solved to determine the optimum number of columns, the length of each column, and the column distribution in different zones. Significant improvement in the recovery of p-xylene using less solvent was obtained when a suitable column length and appropriate distribution of columns in various zones was chosen. Optimization study was also extended for the Varicol operation mode, which is based on a nonsynchronous switching, contrary to the synchronous switching used in traditional SMB systems. It was observed that the performance of the Varicol process is much superior to that of a SMB process, in terms of recovering more p-xylene while consuming less eluent. Results are presented and discussed in detail. The optimum results are also explained using equilibrium theory by locating them in the pure separation region.

Multiobjective Placement of Electronic Components Using Evolutionary Algorithms

The optimal placement of electronic components on a printed circuit board is a well-studied optimization task. However, despite the involvement of multiple conflicting objectives, researchers have mainly used a single objective of minimizing the overall wire length or minimizing the overall heat generation or minimizing the overall time delay in its functioning. In this paper, the problem is treated as a two-objective optimization problem of minimizing the overall wire length and minimizing the failure-rate of the board arising due to uneven local heat accumulation. The proposed strategy uses a novel representation procedure and a multiobjective evolutionary algorithm capable of finding multiple Pareto-optimal solutions simultaneously. Moreover, the flexibility and efficacy of the proposed strategy have been demonstrated by simultaneously optimizing the placement of components and the layout of the board. The convergence and the extent of spread obtained in the solutions reliably by repetitive applications of the proposed procedure should encourage further application of the approach to more complex placement design problems.

Two-objective on-line optimization of supervisory control strategy

The set points of supervisory control strategy are optimized with respect to energy use and thermal comfort for existing HVAC systems. The set point values of zone temperatures, supply duct static pressure, and supply air temperature are the problem variables, while energy use and thermal comfort are the objective functions. The HVAC system model includes all the individual component models developed and validated against the monitored data of an existing VAV system. It serves to calculate energy use during the optimization process, whereas the actual energy use is determined by using monitoring data and the appropriate validated component models. A comparison, done for one summer week, of actual and optimal energy use shows that the on-line implementation of a genetic algorithm optimization program to determine the optimal set points of supervisory control strategy could save energy by 19.5%, while satisfying the minimum zone airflow rates and the thermal comfort. The results also indicate that the application of the two-objective optimization problem can help control daily energy use or daily building thermal comfort, thus saving more energy than the application of the one-objective optimization problem.

パレート最適性を考慮したロバストＩ‐ＰＤ制御系設計法

In this paper, a new design method for robust I-PD controller by using Pareto-optimality is proposed. The proposed design method is formulated as a two-objective optimization problem in which optimization of the settling time and damping ratio is translated into a pole assignment problem. The uncertainties of the plant are represented as a polytope of polynomials, and the computational cost is reduced by using the edge theorem. The genetic algorithm is applied to optimize this problem because of its multiple search property. In order to demonstrate the effectiveness of the proposed design method, we applied the proposed design method to a magnetic levitation system and compared our method with the design method based on the generalized ISE (Integral of Squared Error) criterion.

Multiobjective Sampling Design for Water Distribution Model Calibration

Sampling design (SD) for water distribution systems (WDS) is undoubtedly an important issue, and has been addressed in the past by a number of scientists and practitioners. The aim of the SD methodology developed here is to find a set of optimal network locations at which to place measurement devices. Optimal locations are determined with the aim of collecting data that will be used later on in the calibration of the analyzed WDS hydraulic model. First, existing calibration and SD approaches in the case of WDS are reviewed. After that, SD is formulated as a two-objective optimization problem. The objectives are maximization of the calibrated model accuracy by minimization of the relevant uncertainties, and minimization of total SD costs. The optimal SD problem is then solved using a multiobjective genetic algorithm based on Pareto ranking, niching, and restricted mating. The methodology developed is applied and verified on a case study. At the end, a summary is made and relevant conclusions are drawn.

Multi-objective optimization of an industrial fluidized-bed catalytic cracking unit (FCCU) using genetic algorithm (GA) with the jumping genes operator

The multi-objective optimization of industrial operations using genetic algorithm and its variants, often requires inordinately large amounts of computational (CPU) time. Any adaptation to speed up the solution procedure is, thus, desirable. An adaptation is developed in this study that is inspired from natural genetics. It is based on the concept of jumping genes (JG; transposons). The binary-coded elitist non-dominated sorting genetic algorithm (NSGA-II) is adapted, and the new code, NSGA-II-JG, is used to obtain solutions for the multi-objective optimization of an industrial fluidized-bed catalytic cracking unit (FCCU). This unit is associated with a complex model that is highly compute-intense. The CPU time required for this problem is found to reduce fivefold when NSGA-II-JG is used, as compared to when NSGA-II is used. Solutions of similar two-objective optimization problems for the FCCU are compared. NSGA-II-JG also gives improved convergence characteristics and spread of the optimal Pareto points for two simpler multi-objective optimization problems studied here. Indeed, in one problem, where several optimal Pareto fronts exist, the new code gives the correct, global optimal Pareto set, while the original code (binary-coded NSGA-II) converges to local Paretos. The JG operator is associated with some kind of macro–macro-mutation and introduces higher exploratory capabilities, counteracting the effect of elitism in NSGA-II. We, thus, have a better algorithm incorporating the advantages of elitism. This adaptation can prove to be of considerable value for solving other compute-intense problems in chemical engineering.

Design Of Robust Pole Assignment Based On Pareto‐Optimal Solutions

ABSTRACTIn this paper, a new design method for robust pole assignment based on Pareto‐optimal solutions for an uncertain plant is proposed. The proposed design method is defined as a two‐objective optimization problem in which optimization of the settling time and damping ratio is translated into a pole assignment problem. The uncertainties of the plant are represented as a polytope of polynomials, and the design cost is reduced by using the edge theorem. The genetic algorithm is applied to optimize this problem because of its multiple search property. In order to demonstrate the effectiveness of the proposed design method, we applied the proposed design method to a magnetic levitation system.

A computerized team approach for concurrent product and process design optimization

Concurrent product and process design (CPPD) can be handled by developing a computerized team approach. In the team approach, CPPD at the top-level abstraction is treated as a two-objective optimization problem with a view to two teams: design and manufacture. The design team aims to optimize the overall product functionality (or performance) and the manufacture team to minimize the total manufacturing cost. Satisfaction metrics are used to characterize the degree of how each team prefers a design in seeking the most favorite that best fulfills the goal of a team. Using min and geometric mean operators as a baseline, the satisfaction levels of two teams can be aggregated with respect to strategic team paradigms derived from game theory. With satisfaction metrics, three dual-team based CPPD models are developed, along with the computational algorithms, in an attempt to reveal team activities in the context of design. The procedures to simulate and implement various team design models are demonstrated through a design example.

IMMUNE NETWORK SIMULATION WITH MULTIOBJECTIVE GENETIC ALGORITHMS FOR MULTIDISCIPLINARY DESIGN OPTIMIZATION

The paper presents a method called MOGA-INS for Multidisciplinary Design Optimization (MDO) of systems that involve multiple competing objectives with a mix of continuous and discrete variables. The method is based on the Immune Network Simulation ( INS) approach that has been extended by combining it with a Multi-Objective Genetic Algorithm ( MOGA). MOGA obtains Pareto solutions for multiple objective optimization problems in an all-at-once manner. INS provides a coordination strategy for subsystems in MDO to interact and is naturally suited for genetic algorithm-based optimization methods. The MOGA-INS method is demonstrated with a speed-reducer example, formulated as a two-level two-objective design optimization problem.