Modelling the Pareto-optimal set using B-spline basis functions for continuous multi-objective optimization problems

In the past few years, multi-objective optimization algorithms have been extensively applied in several fields including engineering design problems. A major reason is the advancement of evolutionary multi-objective optimization (EMO) algorithms that are able to find a set of non-dominated points spread on the respective Pareto-optimal front in a single simulation. Besides just finding a set of Pareto-optimal solutions, one is often interested in capturing knowledge about the variation of variable values over the Pareto-optimal front. Recent innovization approaches for knowledge discovery from Pareto-optimal solutions remain as a major activity in this direction. In this article, a different data-fitting approach for continuous parameterization of the Pareto-optimal front is presented. Cubic B-spline basis functions are used for fitting the data returned by an EMO procedure in a continuous variable space. No prior knowledge about the order in the data is assumed. An automatic procedure for detecting gaps in the Pareto-optimal front is also implemented. The algorithm takes points returned by the EMO as input and returns the control points of the B-spline manifold representing the Pareto-optimal set. Results for several standard and engineering, bi-objective and tri-objective optimization problems demonstrate the usefulness of the proposed procedure.     

Customized evolutionary optimization procedure for generating minimum weight compliant mechanisms

In this article, a customized evolutionary optimization procedure is developed for generating minimum weight compliant mechanisms. A previously-suggested concept of multi-objectivization in which a helper objective is introduced in addition to the primary objective of the original single-objective optimization problem (SOOP) is used here. The helper objective is chosen in a way such that it is in conflict with the primary objective, thereby causing an evolutionary multi-objective optimization algorithm to maintain diversity in its population from one generation to another. The elitist non-dominated sorting genetic algorithm (NSGA-II) is customized with a domain-specific initialization strategy, a domain-specific crossover operator, and a domain-specific solution repairing strategy. To make the search process computationally tractable, the proposed methodology is made suitable for parallel computing. A local search methodology is applied on the evolved non-dominated solutions found by the above-mentioned modified NSGA-II to refine the solutions further. Two case studies for tracing curvilinear and straight-line paths are performed. Results demonstrate that solutions having smaller weight than the reference design solution obtained by SOOP are found by the proposed procedure. Interesting facts and observations brought out by the study are also narrated and conclusions of the study are made.     

Multidisciplinary design and optimization of an air launched satellite launch vehicle using a hybrid heuristic search algorithm

A multidisciplinary design and optimization strategy for a multistage air launched satellite launch vehicle comprising of a solid propulsion system to low earth orbit with the implementation of a hybrid heuristic search algorithm is proposed in this article. The proposed approach integrated the search properties of a genetic algorithm and simulated annealing, thus achieving an optimal solution while satisfying the design objectives and performance constraints. The genetic algorithm identified the feasible region of solutions and simulated annealing exploited the identified feasible region in search of optimality. The proposed methodology coupled with design space reduction allows the designer to explore promising regions of optimality. Modules for mass properties, propulsion characteristics, aerodynamics, and flight dynamics are integrated to produce a high-fidelity model of the vehicle. The objective of this article is to develop a design strategy that more efficiently and effectively facilitates multidisciplinary design analysis and optimization for an air launched satellite launch vehicle.     

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.     

Practical optimization of group piles using discrete Lagrange multiplier method

This paper presents a practical methodology for designing group piles of a bridge foundation that can minimize the construction cost. The constraint functions for the designed pile group are constructed according to the design code currently adopted in local engineering practice. A new procedure based on the discrete Lagrange multiplier (DLM) method is proposed for searching the optimal solution. Seven real-world design cases are used to test the validity and the performance of the proposed procedure and algorithm, in which the DLM solutions are compared with the global optimum solutions obtained by the exhaustive searching method (ESM). Vast improvement in the computational efficiency is achieved, as the DLM method can find local minimum solutions in less than 2.0 to 4.0 minutes whereas the time required with the ESM is 10,000 to 25,000 times longer. In the case studies presented, the construction costs in conjunction with the use of the DLM method differ from the global minimum costs by less than 6.0%, but the savings over the original designs can be as high as 13% to 56%.     

Optimization of machining datum selection and machining tolerance allocation with genetic algorithms

In machining process planning, selection of machining datum and allocation of machining tolerances are crucial as they directly affect the part quality and machining efficiency. This study explores the feasibility to build a mathematical model for computer aided process planning (CAPP) to find the optimal machining datum set and machining tolerances simultaneously for rotational parts. Tolerance chart and an efficient dimension chain tracing method are utilized to establish the relationship between machining datums and tolerances. A mixed-discrete nonlinear optimization model is formulated with the manufacturing cost as the objective function and blueprint tolerances and machine tool capabilities as constraints. A directed random search method, genetic algorithm (GA), is used to find optimum solutions. The computational results indicate that the proposed methodology is capable and robust in finding the optimal machining datum set and tolerances. The proposed model and solution procedure can be used as a building block for computer automated process planning.     

Towards automating the discovery of certain innovative design principles through a clustering-based optimization technique

In this article, a methodology is proposed for automatically extracting innovative design principles which make a system or process (subject to conflicting objectives) optimal using its Pareto-optimal dataset. Such ‘higher knowledge’ would not only help designers to execute the system better, but also enable them to predict how changes in one variable would affect other variables if the system has to retain its optimal behaviour. This in turn would help solve other similar systems with different parameter settings easily without the need to perform a fresh optimization task. The proposed methodology uses a clustering-based optimization technique and is capable of discovering hidden functional relationships between the variables, objective and constraint functions and any other function that the designer wishes to include as a ‘basis function’. A number of engineering design problems are considered for which the mathematical structure of these explicit relationships exists and has been revealed by a previous study. A comparison with the multivariate adaptive regression splines (MARS) approach reveals the practicality of the proposed approach due to its ability to find meaningful design principles. The success of this procedure for automated innovization is highly encouraging and indicates its suitability for further development in tackling more complex design scenarios.     

Optimization of transit network layout and headway with a combined genetic algorithm and simulated annealing method

This article proposes a methodology for optimizing transit networks, including both route structures and headways. Given information on transit demand, transit fleet size and street network in the transit service area, the methodology seeks to minimize transfers and total user cost while maximizing service coverage. The goal is to provide an effective mathematical solution procedure with minimal reliance on heuristics to solve large-scale transit network optimization problems. This article describes the representation of the transit route network and the associated network search spaces, the representation of route network headways and the associated search spaces, the total user cost objective functions, and a stochastic global search scheme based on a combined genetic algorithm and simulated annealing search method. The methodology has been tested with published benchmark problems and applied to a large-scale realistic network optimization problem. The results show that the methodology is capable of producing improved solutions to large-scale transit network design problems.     

The benefits of adaptive parametrization in multi-objective Tabu Search optimization

In real-world optimization problems, large design spaces and conflicting objectives are often combined with a large number of constraints, resulting in a highly multi-modal, challenging, fragmented landscape. The local search at the heart of Tabu Search, while being one of its strengths in highly constrained optimization problems, requires a large number of evaluations per optimization step. In this work, a modification of the pattern search algorithm is proposed: this modification, based on a Principal Components’ Analysis of the approximation set, allows both a re-alignment of the search directions, thereby creating a more effective parametrization, and also an informed reduction of the size of the design space itself. These changes make the optimization process more computationally efficient and more effective – higher quality solutions are identified in fewer iterations. These advantages are demonstrated on a number of standard analytical test functions (from the ZDT and DTLZ families) and on a real-world problem (the optimization of an axial compressor preliminary design).     

Automatic segmentation system for liver tumors based on the multilevel thresholding and electromagnetism optimization algorithm

In this article, we propose an automated segmentation system for liver tumors using magnetic resonance imaging and computed tomography. The proposed system is based on the algorithm of multilevel thresholding with electromagnetism optimization (EMO). The system starts with visualizing a patient's digital communication in medicine (DICOM) abdominal data set in three views. Two-stage active contour segmentation methods that integrate region-based local and global techniques using the active geodesic contour technique are proposed to segment the liver. To increase the accuracy and speed of segmentation for liver images, we identify the optimal threshold of the image segmentation method based on EMO with Otsu and Kapur algorithms. EMO offers interesting search capabilities while keeping a low computational cost. The proposed system was tested using a set of five DICOM data sets. All images were of the same size and stored in JPEG format (512 × 512 pixels). Experimental results illustrate that the proposed system outperforms state-of-the-art methods such as the watershed algorithm. The average sensitivity, specificity, and accuracy of the segmented liver using the active contour model were 97.05%, 99.88%, and 98.47%, respectively. Moreover, the average sensitivity, specificity, and accuracy of the segmented liver tumor results were 94.15%, 99.57%, and 96.86%, respectively.     

Design of a decentralized framework for collaborative product design using memetic algorithms

This paper proposes a generalized bi-level decentralized framework to model collaborative design problems over autonomous stakeholders with each having different objectives. At the system level, a system solution derived from the Pareto concept is created. A facilitator agent is introduced to search for Pareto optimal solutions based on a Memetic Algorithm (MA). At the design disciplinary level, design agents representing design teams are introduced to optimize their own objectives. The proposed framework will guide the collaborative designers to converge to Pareto optimal solutions given any forms of design utility functions. The only information exchanged between the two levels is numerical values instead of utility functions. Therefore sensitive (private) design information can be protected. Three comparison experiments are conducted to evaluate the solution quality and explore the applicability of the proposed framework to collaborative design problems.     

The harmony search heuristic algorithm for discrete structural optimization

Many methods have been developed and are in use for structural size optimization problems, in which the cross-sectional areas or sizing variables are usually assumed to be continuous. In most practical structural engineering design problems, however, the design variables are discrete. This paper proposes an efficient optimization method for structures with discrete-sized variables based on the harmony search (HS) heuristic algorithm. The recently developed HS algorithm was conceptualized using the musical process of searching for a perfect state of harmony. It uses a stochastic random search instead of a gradient search so that derivative information is unnecessary. In this article, a discrete search strategy using the HS algorithm is presented in detail and its effectiveness and robustness, as compared to current discrete optimization methods, are demonstrated through several standard truss examples. The numerical results reveal that the proposed method is a powerful search and design optimization tool for structures with discrete-sized members, and may yield better solutions than those obtained using current methods.     

Multi-objective parametric optimization of powder mixed electro-discharge machining using response surface methodology and non-dominated sorting genetic algorithm

Powder mixed electro-discharge machining (EDM) is being widely used in modern metal working industry for producing complex cavities in dies and moulds which are otherwise difficult to create by conventional machining route. It has been experimentally demonstrated that the presence of suspended particle in dielectric fluid significantly increases the surface finish and machining efficiency of EDM process. Concentration of powder (silicon) in the dielectric fluid, pulse on time, duty cycle, and peak current are taken as independent variables on which the machining performance was analysed in terms of material removal rate (MRR) and surface roughness (SR). Experiments have been conducted on an EZNC fuzzy logic Die Sinking EDM machine manufactured by Electronica Machine Tools Ltd. India. A copper electrode having diameter of 25?mm is used to cut EN 31 steel for one hour in each trial. Response surface methodology (RSM) is adopted to study the effect of independent variables on responses and develop predictive models. It is desired to obtain optimal parameter setting that aims at decreasing surface roughness along with larger material removal rate. Since the responses are conflicting in nature, it is difficult to obtain a single combination of cutting parameters satisfying both the objectives in any one solution. Therefore, it is essential to explore the optimization landscape to generate the set of dominant solutions. Non-sorted genetic algorithm (NSGA) has been adopted to optimize the responses such that a set of mutually dominant solutions are found over a wide range of machining parameters.     

Variation-aware clock network buffer sizing using robust multi-objective optimization

Many engineering optimization problems include unavoidable uncertainties in parameters or variables. Ignoring such uncertainties when solving the optimization problems may lead to inferior solutions that may even violate problem constraints. Another challenge in most engineering optimization problems is having different conflicting objectives that cannot be minimized simultaneously. Finding a balanced trade-off between these objectives is a complex and time-consuming task. In this paper, an optimization framework is proposed to address both of these challenges. First, we exploit a self-calibrating multi-objective framework to achieve a balanced trade-off between the conflicting objectives. Then, we develop the robust counterpart of the uncertainty-aware self-calibrating multi-objective optimization framework. The significance of this framework is that it does not need any manual tuning by the designer. We also develop a mathematical demonstration of the objective scale invariance property of the proposed framework. The engineering problem considered in this paper to illustrate the effectiveness of the proposed framework is a popular sizing problem in digital integrated circuit design. However, the proposed framework can be applied to any uncertain multi-objective optimization problem that can be formulated in the geometric programming format. We propose to consider variations in the sizes of circuit elements during the optimization process by employing ellipsoidal uncertainty model. For validation, several industrial clock networks are sized by the proposed framework. The results show a significant reduction in one objective (power, on average 38 %) as well as significant increase in the robustness of solutions to the variations. This is achieved with no significant degradation in the other objective (timing metrics of the circuit) or reduction in its standard deviation which demonstrates a more robust solution.     

Explicit consideration of multiple objectives in cellular manufacturing

Although many methodologies have been proposed for solving the cell-formation problem, few of them explicitly consider the existence of multiple objectives in the design process. In this article, the development of multi-objective genetic programming single-linkage cluster analysis (GP-SLCA), an evolutionary methodology for the solution of the multi-objective cell-formation problem, is described. The proposed methodology combines an existing algorithm for the solution of single-objective cell-formation problems with NSGA-II, an elitist evolutionary multi-objective optimization technique. Multi-objective GP-SLCA is able to generate automatically a set of non-dominated solutions for a given multi-objective cell-formation problem. The benefits of the proposed approach are illustrated using an example test problem taken from the literature and an industrial case study.     

The constraints satisfaction problem approach in the design of an architectural functional layout

A design support system with a new strategy for finding the optimal functional configurations of rooms for architectural layouts is presented. A set of configurations satisfying given constraints is generated and ranked according to multiple objectives. The method can be applied to problems in architectural practice, urban or graphic design—wherever allocation of related geometrical elements of known shape is optimized. Although the methodology is shown using simplified examples—a single story residential building with two apartments each having two rooms—the results resemble realistic functional layouts. One example of a practical size problem of a layout of three apartments with a total of 20 rooms is demonstrated, where the generated solution can be used as a base for a realistic architectural blueprint. The discretization of design space is discussed, followed by application of a backtrack search algorithm used for generating a set of potentially ‘good’ room configurations. Next the solutions are classified by a machine learning method (FFN) as ‘proper’ or ‘improper’ according to the internal communication criteria. Examples of interactive ranking of the ‘proper’ configurations according to multiple criteria and choosing ‘the best’ ones are presented. The proposed framework is general and universal—the criteria, parameters and weights can be individually defined by a user and the search algorithm can be adjusted to a specific problem.     

Solving kinematics problems by efficient interval partitioning

In this study, a generic solution methodology is presented for solving constraint satisfaction problems with smooth continuous constraint functions. We apply the proposed method in solving small scale kinematics problems with the goal of identifying all real solutions to a given problem. The developed approach assumes a collaborative methodology that integrates interval partitioning, a new interval inference method, and local search. The resulting methodology aims at reducing the search space and discarding infeasible sub-spaces effectively and reliably during the early stages of the search so that exact solutions can be identified faster by local methods.     

Cold-standby redundancy optimization for nonrepairable systems

A solution methodology is described and demonstrated to determine optimal design configurations for nonrepairable series-parallel systems with cold-standby redundancy. This problem formulation considers non-constant component hazard functions and imperfect switching. The objective of the redundancy allocation problem is to select from available components and to determine an optimal design configuration to maximize system reliability. For cold-standby redundancy, other formulations have generally required exponential component time-to-failure and perfect switching assumptions. For this paper, there are multiple component choices available for each subsystem and component time-to-failure is distributed according to an Erlang distribution. Optimal solutions are determined based on an equivalent problem formulation and integer programming. Compared to other available algorithms, the methodology presented here more accurately models many engineering design problems with cold-standby redundancy. Previously, it has been difficult to determine optimal solutions for this class of problems or even lo efficiently calculate system reliability. The methodology is successfully demonstrated on a large problem with 14 subsystems.     

Topological and parametric optimization of gear trains

A method for automating the design of gear trains comprised of simple, compound, bevel and worm is described. The search process combines topological changes, discrete variable choices and continuous variable optimization. By combing best-first search, implicit enumeration, automated optimization invocation and gradient-based optimization, a near guarantee of the optimal solution can be made. While the combination of methods is specific to gear trains, there are aspects of the work that make it amenable to other engineering design problems. In addition, the topological and discrete modifications to the candidate solutions are specific to gear trains, but the graph grammar methodology that is adopted has been tailored to other problems. This article presents details on the rules that generate feasible gear trains, the evaluation routines used in determining the objective functions and constraints, and the interaction among the three search methods. Resulting gear trains are presented for a variety of gear problems.