Uncertainty analysis of trade‐offs between multiple responses using hypervolume

When multiple responses are considered in process optimization, the degree to which they can be simultaneously optimized depends on the optimization objectives and the amount of trade‐offs between the responses. The normalized hypervolume of the Pareto front is a useful summary to quantify the amount of trade‐offs required to balance performance across the multiple responses. To quantify the impact of uncertainty of the estimated response surfaces and add realism to what future data to expect, 2 versions of the scaled normalized hypervolume of the Pareto front are presented. To demonstrate the variation of the hypervolume distributions, we explore a case study for a chemical process involving 3 responses, each with a different type of optimization goal. Results show that the global normalized hypervolume characterizes the proximity to the ideal results possible, while the instance‐specific summary considers the richness of the front and the severity of trade‐offs between alternatives. The 2 scaling schemes complement each other and highlight different features of the Pareto front and hence are useful to quantify what solutions are possible for simultaneous optimization of multiple responses.     

Balancing Multiple Criteria Incorporating Cost using Pareto Front Optimization for Split‐Plot Designed Experiments

Finding a D‐optimal design for a split‐plot experiment requires knowledge of the relative size of the whole plot (WP) and sub‐plot error variances. Since this information is typically not known a priori, we propose an optimization strategy based on balancing performance across a range of plausible variance ratios. This approach provides protection against selecting a design which could be sub‐optimal if a single initial guess is incorrect. In addition, options for incorporating experimental cost into design selection are explored. The method uses Pareto front multiple criteria optimization to balance these objectives and allows the experimenter to understand the trade‐offs between several design choices and select one that best suits the goals of the experiment. We present new algorithms for populating the Pareto front for the split‐plot situation when the number of WPs is either fixed or flexible. We illustrate the method with a case study and demonstrate how considering robustness across variance ratios offers improved performance. The Pareto approach identifies multiple promising designs, and allows the experimenter to understand trade‐offs between alternatives and examining their robustness to different ways of combining the objectives. New graphical summaries for up to four criteria are developed to help guide improved decision‐making. Copyright © 2012 John Wiley & Sons, Ltd.     

MONSS: A multi-objective nonlinear simplex search approach

This article presents a novel methodology for dealing with continuous box-constrained multi-objective optimization problems (MOPs). The proposed algorithm adopts a nonlinear simplex search scheme in order to obtain multiple elements of the Pareto optimal set. The search is directed by a well-distributed set of weight vectors, each of which defines a scalarization problem that is solved by deforming a simplex according to the movements described by Nelder and Mead's method. Considering an MOP with n decision variables, the simplex is constructed using n+1 solutions which minimize different scalarization problems defined by n+1 neighbor weight vectors. All solutions found in the search are used to update a set of solutions considered to be the minima for each separate problem. In this way, the proposed algorithm collectively obtains multiple trade-offs among the different conflicting objectives, while maintaining a proper representation of the Pareto optimal front. In this article, it is shown that a well-designed strategy using just mathematical programming techniques can be competitive with respect to the state-of-the-art multi-objective evolutionary algorithms against which it was compared.     

Multiobjective optimization with an adaptive weight determination scheme using the concept of hyperplane

Finding an optimum design that satisfies all performances in a design problem is very challenging. To overcome this problem, multiobjective optimization methods have been researched to obtain Pareto optimum solutions. Among the different methods, the weighted sum method is widely used for its convenience. However, since the different weights do not always guarantee evenly distributed solutions on the Pareto front, the weights need to be determined systematically. Therefore, this paper presents a multiobjective optimization using a new adaptive weight determination scheme. Solutions on the Pareto front are gradually found with different weights, and the values of these weights are adaptively determined by using information from the previously obtained solutions' positions. For an n-objective problem, a hyperplane is constructed in n -dimensional space, and new weights are calculated to find the next solutions. To confirm the effectiveness of the proposed method, benchmarking problems that have different types of Pareto front are tested, and a topology optimization problem is performed as an engineering problem. A hypervolume indicator is used to quantitatively evaluate the proposed method, and it is confirmed that optimized solutions that are evenly distributed on the Pareto front can be obtained by using the proposed method.     

An optimization model of sewage rehabilitation

Abstract

Due to low visibility, sewer systems are difficult to monitor, maintain and rehabilitate. To prevent failures, environmental pollution, and wastewater treatment overflow, regular rehabilitation of sewage is necessary. However, sewage rehabilitation usually costs an immense amount of money and is hampered by a limited budget. Thus, efficient planning of maintenance and rehabilitation for sewage upkeep is demanded. In this paper, an optimization model has been built to find an appropriate rehabilitation strategy consisting of a rehabilitation method and a substitute material for each pipe failure under a limited budget. The optimization model was designed to search for a Pareto curve (or trade‐off front) consisting of a set of optimal solutions with desirable rehabilitation effectiveness at the least cost. This paper employs genetic algorithms (GA) to obtain a Pareto curve at a low computation cost for large and complex sewer systems. This optimization model was applied to a sewer system in the 15^th district of Kaohsiung City, Taiwan. Compared with the experts’ manual estimation, the optimization model saved about 20% of the rehabilitation cost for Kaohsiung City.     

Non-dominated archiving multi-colony ant algorithm for multi-objective optimization: Application to multi-purpose reservoir operation

Multi-objective optimization using heuristic methods has been established as a subdiscipline that combines the fields of heuristic computation and classical multiple criteria decision making. This article presents the Non-dominated Archiving Ant Colony Optimization (NA-ACO), which benefits from the concept of a multi-colony ant algorithm and incorporates a new information-exchange policy. In the proposed information-exchange policy, after a given number of iterations, different colonies exchange information on the assigned objective, resulting in a set of non-dominated solutions. The non-dominated solutions are moved into an offline archive for further pheromone updating. Performance of the NA-ACO is tested employing two well-known mathematical multi-objective benchmark problems. The results are promising and compare well with those of well-known NSGA-II algorithms used in real-world multi-objective-optimization problems. In addition, the optimization of reservoir operating policy with multiple objectives (i.e. flood control, hydropower generation and irrigation water supply) is considered and the associated Pareto front generated.     

Dual-mode nested search method for categorical uncertain multi-objective optimization

Categorical multi-objective optimization is an important issue involved in many matching design problems. Non-numerical variables and their uncertainty are the major challenges of such optimizations. Therefore, this article proposes a dual-mode nested search (DMNS) method. In the outer layer, kriging metamodels are established using standard regular simplex mapping (SRSM) from categorical candidates to numerical values. Assisted by the metamodels, a k-cluster-based intelligent sampling strategy is developed to search Pareto frontier points. The inner layer uses an interval number method to model the uncertainty of categorical candidates. To improve the efficiency, a multi-feature convergent optimization via most-promising-area stochastic search (MFCOMPASS) is proposed to determine the bounds of objectives. Finally, typical numerical examples are employed to demonstrate the effectiveness of the proposed DMNS method.     

On the Pareto optimum sensitivity analysis in multicriteria optimization

To analyse the trade‐off relations among the set of criteria in multicriteria optimization, Pareto optimum sensitivity analysis is systematically studied in this paper. Original contributions cover two parts: theoretical demonstrations are firstly made to validate the gradient projection method in Pareto optimum sensitivity analysis. It is shown that the projected gradient direction evaluated at a given Pareto optimum in the design variable space rigorously corresponds to the tangent direction of the Pareto curve/surface at that point in the objective space. This statement holds even for the change of the set of active constraints in the perturbed problem. Secondly, a new active constraint updating strategy is proposed, which permits the identification of the active constraint set change, to determine the influence of this change upon the differentiability of the Pareto curve and finally to compute directional derivatives in non‐differentiable cases. This work will highlight some basic issues in multicriteria optimization. Some numerical problems are solved to illustrate these novelties. Copyright © 2003 John Wiley & Sons, Ltd.     

Pareto archived dynamically dimensioned search with hypervolume-based selection for multi-objective optimization

Pareto archived dynamically dimensioned search (PA-DDS) is a parsimonious multi-objective optimization algorithm with only one parameter to diminish the user's effort for fine-tuning algorithm parameters. This study demonstrates that hypervolume contribution (HVC) is a very effective selection metric for PA-DDS and Monte Carlo sampling-based HVC is very effective for higher dimensional problems (five objectives in this study). PA-DDS with HVC performs comparably to algorithms commonly applied to water resources problems (?-NSGAII and AMALGAM under recommended parameter values). Comparisons on the CEC09 competition show that with sufficient computational budget, PA-DDS with HVC performs comparably to 13 benchmark algorithms and shows improved relative performance as the number of objectives increases. Lastly, it is empirically demonstrated that the total optimization runtime of PA-DDS with HVC is dominated (90% or higher) by solution evaluation runtime whenever evaluation exceeds 10 seconds/solution. Therefore, optimization algorithm runtime associated with the unbounded archive of PA-DDS is negligible in solving computationally intensive problems.     

Inverse modelling of multi-objective thermodynamically optimized turbojet engines using GMDH-type neural networks and evolutionary algorithms

A novel approach is presented in this article for obtaining inverse mapping of thermodynamically Pareto-optimized ideal turbojet engines using group method of data handling (GMDH)-type neural networks and evolutionary algorithms (EAs). EAs are used in two different aspects. Firstly, multi-objective EAs (non–dominated sorting genetic algorithm-II) with a new diversity preserving mechanism are used for Pareto-based optimization of the thermodynamic cycle of ideal turbojet engines considering four important conflicting thermodynamic objectives, namely, specific thrust ({ST}), specific fuel consumption ({SFC}), propulsive efficiency (η_p), and thermal efficiency (η_t). The best obtained Pareto front, as a result, is a data table representing data pairs of non-dominated vectors of design variables, which are Mach number and pressure ratio, and the corresponding four objective functions. Secondly, EAs and singular value decomposition are deployed simultaneously for optimal design of both connectivity configuration and the values of coefficients, respectively, involved in GMDH-type neural networks which are used for the inverse modelling of the input–output data table obtained as the best Pareto front. Therefore, two different polynomial relations among the four thermo-mechanical objectives and both Mach number and pressure ratio are searched using that Pareto front. The results obtained in this paper are very promising and show that such important relationships may exist and could be discovered using both multi-objective EAs and evolutionarily designed GMDH-type neural networks.     

A TABOO SEARCH BASED APPROACH TO FIND THE PARETO OPTIMAL SET IN MULTIPLE OBJECTIVE OPTIMIZATION

Taboo search is a heuristic optimization technique which works with a neighbourhood of solutions to optimize a given objective function. It is generally applied to single objective optimization problems. Taboo search has the potential for solving multiple objective optimization (MOO) problems, because it works with more than one solution at a time, and this gives it the opportunity to evaluate multiple objective functions simultaneously. In this paper, a taboo search based algorithm is developed to find Pareto optimal solutions in multiple objective optimization problems. The developed algorithm has been tested with a number of problems and compared with other techniques. Results obtained from this work have proved that a taboo search based algorithm can find Pareto optimal solutions in MOO effectively.     

Optimal power system generation scheduling by multi-objective genetic algorithms with preferences

Power system generation scheduling is an important issue both from the economical and environmental safety viewpoints. The scheduling involves decisions with regards to the units start-up and shut-down times and to the assignment of the load demands to the committed generating units for minimizing the system operation costs and the emission of atmospheric pollutants.As many other real-world engineering problems, power system generation scheduling involves multiple, conflicting optimization criteria for which there exists no single best solution with respect to all criteria considered. Multi-objective optimization algorithms, based on the principle of Pareto optimality, can then be designed to search for the set of nondominated scheduling solutions from which the decision-maker (DM) must a posteriori choose the preferred alternative. On the other hand, often, information is available a priori regarding the preference values of the DM with respect to the objectives. When possible, it is important to exploit this information during the search so as to focus it on the region of preference of the Pareto-optimal set.In this paper, ways are explored to use this preference information for driving a multi-objective genetic algorithm towards the preferential region of the Pareto-optimal front. Two methods are considered: the first one extends the concept of Pareto dominance by biasing the chromosome replacement step of the algorithm by means of numerical weights that express the DM’ s preferences; the second one drives the search algorithm by changing the shape of the dominance region according to linear trade-off functions specified by the DM.The effectiveness of the proposed approaches is first compared on a case study of literature. Then, a nonlinear, constrained, two-objective power generation scheduling problem is effectively tackled.     

基于改进NSGA-Ⅱ算法的电-气-热综合能源系统多目标优化

随着电力、天然气和热力网络耦合紧密程度不断加深,综合能源系统协同优化成为了新的研究热点。提出一种适用于含非凸约束条件的综合能源系统多目标优化问题的改进NSGA-Ⅱ算法,通过维护全局的帕累托最优解集提升解的搜索效率,同时采用动态调整法,提高在高维等式约束下找到可行解的概率。算例分析验证了该方法的有效性。     

Uncertainty‐based robust aerodynamic optimization of rotor blades

The aerodynamic performance of a compressor is highly sensitive to uncertain working conditions. This paper presents an efficient robust aerodynamic optimization method on the basis of nondeterministic computational fluid dynamic (CFD) simulation and multi‐objective genetic algorithm (MOGA). A nonintrusive polynomial chaos method is used in conjunction with an existing well‐verified CFD module to quantify the uncertainty propagation in the flow field. This method is validated by comparing with a Monte Carlo method through full 3D CFD simulations on an axial compressor (National Aeronautics and Space Administration rotor 37). On the basis of the validation, the nondeterministic CFD is coupled with a surrogate‐based MOGA to search for the Pareto front. A practical engineering application is implemented to the robust aerodynamic optimization of rotor 37 under random outlet static pressure. Two curve angles and two sweep angles at tip and hub are used as design variables. Convergence analysis shows that the surrogate‐based MOGA can obtain the Pareto front properly. Significant improvements of both mean and variance of the efficiency are achieved by the robust optimization. The comparison of the robust optimization results with that of the initial design, and a deterministic optimization demonstrate that the proposed method can be applied to turbomachinery successfully. Copyright © 2012 John Wiley & Sons, Ltd.     

Multicriteria airport gate assignment and Pareto simulated annealing

This paper addresses an airport gate assignment problem with multiple objectives. The objectives are to minimize the number of ungated flights and the total passenger walking distances or connection times as well as to maximize the total gate assignment preferences. The problem examined is an integer program with multiple objectives (one of them being quadratic) and quadratic constraints. Of course, such a problem is inherently difficult to solve. We tackle the problem by Pareto simulated annealing in order to get a representative approximation for the Pareto front. Results of computational experiments are presented. To the best of our knowledge, this is the first attempt to consider the airport gate assignment problem with multiple objectives.     

A non‐dominance‐based online stopping criterion for multi‐objective evolutionary algorithms

A non‐dominance criterion‐based metric that tracks the growth of an archive of non‐dominated solutions over a few generations is proposed to generate a convergence curve for multi‐objective evolutionary algorithms (MOEAs). It was observed that, similar to single‐objective optimization problems, there were significant advances toward the Pareto optimal front in the early phase of evolution while relatively smaller improvements were obtained as the population matured. This convergence curve was used to terminate the MOEA search to obtain a good trade‐off between the computational cost and the quality of the solutions. Two analytical and two crashworthiness optimization problems were used to demonstrate the practical utility of the proposed metric. Copyright © 2010 John Wiley & Sons, Ltd.     

Computational strategy for multiobjective optimization of composite stiffened panels

This paper presents a multiobjective optimization methodology for composite stiffened panels. The purpose is to improve the performances of an existing design of stiffened composite panels in terms of both its first buckling load and ultimate collapse or failure loads. The design variables are the stacking sequences of the skin and of the stiffeners of the panel. The optimization is performed using a multiobjective evolutionary algorithm specifically developed for the design of laminated parts. The algorithm takes into account the industrial design guidelines for stacking sequence design. An original method is proposed for the initialization of the optimization that significantly accelerates the search for the Pareto front. In order to reduce the calculation time, Radial Basis Functions under Tension are used to approximate the objective functions. Special attention is paid to generalization errors around the optimum. The multiobjective optimization results in a wide set of trade-offs, offering important improvements for both considered objectives, among which the designer can make a choice.     

Shape design of internal cooling passages within a turbine blade

The article concerns the optimization of the shape and location of non-circular passages cooling the blade of a gas turbine. To model the shape, four Bezier curves which form a closed profile of the passage were used. In order to match the shape of the passage to the blade profile, a technique was put forward to copy and scale the profile fragments into the component, and build the outline of the passage on the basis of them. For so-defined cooling passages, optimization calculations were carried out with a view to finding their optimal shape and location in terms of the assumed objectives. The task was solved as a multi-objective problem with the use of the Pareto method, for a cooling system composed of four and five passages. The tool employed for the optimization was the evolutionary algorithm. The article presents the impact of the population on the task convergence, and discusses the impact of different optimization objectives on the Pareto optimal solutions obtained. Due to the problem of different impacts of individual objectives on the position of the solution front which was noticed during the calculations, a two-step optimization procedure was introduced. Also, comparative optimization calculations for the scalar objective function were carried out and set up against the non-dominated solutions obtained in the Pareto approach. The optimization process resulted in a configuration of the cooling system that allows a significant reduction in the temperature of the blade and its thermal stress.     

A variant space search method and its application to the optimization design of compressor profiles

Optimization methods have been widely used in practical engineering, with search efficiency and global search ability being the main evaluation criteria. In this article, the Bezier curve equivalent recursion is used in a genetic algorithm (GA) to realize the variant space search to improve the search efficiency and global search ability. The parameters related to this method are investigated by an optimization test of the simple curve approximation, which is then used for optimization designs of supersonic and transonic profiles. The results show that the GA can be improved if the variant space search method is added.