Glossary

Accuracy

The accuracy measures the capability of the optimization algorithm to find the function's extreme.

Algorithm

A precise sequence of actions specifying how to solve a problem.

Constraint

Constraints are quantities imposed to the project that keep values confined within prescribed bounds. These bounds represent restrictions and limits that the designer must meet due to norms, functionalities.

Correlation

The correlation coefficient is a measure of the closeness of the linear relationship between two variables. Positive values of the correlation indicate a tendency of the two variables to increase together. When the coefficient is negative, large values of the first variable are associated with small values of the second one.

Design Of Experiments (DOE)

Design Of Experiments (DOE) provides a strong and universal framework to design and analyze all comparative experiments. With the One Factor At Time approach (OFAT), many runs are usually needed to get sufficient information and this is generally prohibitive. The DOE approach is in direct contrast to OFAT because it considers all factors simultaneously. With DOE the best variable settings are used for obtaining the most relevant qualitative information from a database of experiments making the smallest possible number of experiments. This method is important because it eliminated redundant observations reducing the time and resources to make the experiments. DOE can be even used to look for robust solutions or to create the training set for the Response Surface Methodology.

Design Space

The design space is the entire range of all independent variables, i.e. measures or amounts that are directly controllable.

Design Variables

The design variables define what can be changed in the model, such as the wall thickness, hole diameter, fillet radius, and so on.

DFSS

Design For Six Sigma (DFSS) is a systematic methodology utilizing tools, training and measurements to enable to design products and processes that meet customer expectations and can be produced at Six Sigma quality levels.

Distribution

A distribution is given by set of numbers and their frequency of occurrence collected from measurements over a statistical set.

Extrapolation

The extrapolation is the estimation of a value outside the range of known values. This method requires caution since it assumes that the estimated value follows logically from the known values.

Frequency Chart

The frequency chart summarizes the distribution of a data set. The plot is obtained by splitting the range of the data into equal sized classes and counting the number of points that fall into each class. This type of chart reveals the kind of distribution, where the data is located and how spread out the data is.

Gaussian Distribution

See Normal Distribution.

Genetic algorithm

Genetic algorithms (GA) are algorithms used to find approximate solutions to difficult-to-solve problems. The name for these algorithms is inspired by biological processes of natural selection, mutation, and genetic crossover. At the beginning, an initial population is generated, then all of the individual are evaluated by a fitness function. The generation is sorted with those having better fitness at the top, representing better solutions to the problem. The next step of a GA is to generate a second generation, that is different from the initial generation, using the genetic operators selection, crossover and mutation. The algorithm continues by generating the third, fourth, fifth,... generation, generally increasing the average fitness and until one of the generations contains solutions which are good enough.

Global Optimum

Optimum which corresponds to a minimum in the global region of the design space.

Gradient

The gradient is the vector having coordinate components that are all the partial derivatives of a function with respect to its variables.

Gradient based algorithm

Optimization techniques for non-linear functions which attempt to move incrementally to successively lower points in search space, in order to locate a minimum. These methods make use of the gradient (or the approximate gradient) in order to find the lower points.

History Plot

A history plot is a two dimensional plot for a data series. In the mono-objective, history plot is possible to see that the optimization algorithms evolve the starting points generating designs that have a better value.

Interpolation

Interpolation is the estimation of a value between two known values. Interpolation works by creating a Response Surface (RSM) with two or more given points and then applying this RSM to the required value.

Kurtosis

Kurtosis is a measure of the peakedness of the probability distribution. To evaluate the kurtosis it is necessary to compute the fourth standardized moment. If the kurtosis is positive, the distribution has longer tails than a normal distribution with the same standard deviation.

Local Optimum

Optimum which corresponds to a minimum in the local region of the design space.

Mean

The mean is one of several indexes of central tendency used to indicate the point on the scale of measures where the values is centered. The mean is the average of the values in the set. Numerically, it equals the sum of all values divided by the number of values in the set.

Monte Carlo

In optimization, Monte Carlo methods consist in using random evaluations and to simply picking the best one. There is no way to know if the solution is optimal or not.

Multi-Objectives Optimization

It consists in optimizing multiple objectives to find the best tradeoffs. In multi-objective optimization, there is not one best solution, but instead an infinite number of optimal solutions that lie on the Pareto Frontier. For example there is a tradeoff between driving fast and getting good gas mileage, and once the optimal solutions are found, one needs to pick the best tradeoff for the current situation.

Mutation

Mutation is a reproduction operator for Genetic algorithm designed to maintain diversification in the search. It provides a small amount of random search in the GA process ensuring that all the points in the design space can be reached.

NLPQL

Nonlinear Programming by Quadratic Lagrangian (NLPQL) method builds a quadratic approximation to the Lagrange function and linear approximations to all output constraints at each iteration, starting with the identity matrix for the Hessian of the Lagrangian, and gradually updating it using the BFGS (Broydon-Fletcher-Goldfarb-Shanno) method. On each iteration, a quadratic programming problem is solved to find an improved design until the final convergence to the optimum design.

Normal Distribution

The normal (or Gaussian) distribution is an extremely important probability distribution in many fields. It has the well known symmetrical "bell-shaped" form. The standard normal distribution is the normal distribution with a mean of zero and a standard deviation of one.

Objective

Objectives are the response parameter that the designer wish to maximize or minimize (minimize the cost, maximize the efficiency).

One Factor At Time

The One Factor At Time (OFAT) approach is the traditional approach for planning experiment. It consists on testing one factor at a time. The first decision variable is moved while the other decision variables are held constant, then the next variable is examined, and so on. With OFAT, many experiments are usually needed to get sufficient information and this is generally prohibitive.

Optimization

Automated modifications of the analysis model parameters to achieve a desired objective while satisfying specified design requirements.

Pareto Diagram

A Pareto diagram focuses on efforts or on the problems that have the greatest potential for improvement by showing relative frequency and/or size in a descending bar graph. Based on the proven Pareto principle: 20% of the sources cause 80% of any problems.

Pareto Efficiency

Or Pareto optimality. Given a set of tradeoffs of goals such as performance and cost, improving performance alternative allocations of, say, goods or income for a set of individuals, a movement from one allocation to another that can make at least one individual better off without making any other individual worse off is called a Pareto improvement. An allocation is Pareto efficient or Pareto optimal when no further Pareto improvements can be made.

Pareto Frontier

The plot of the objective functions whose non-dominated vectors are in the Pareto optimal set is called the Pareto Frontier.

Pareto Optimal Solution

A Pareto Optimal Solution is the one where in order to improve one criterion, other criteria must get worse. In product design, the Pareto surface is the mathematical embodiment of all possible optimal solutions. Thus, the mathematics of optimization focuses on finding solutions close to this Pareto surface.

PIDO

Process Integration and Design Optimization (PIDO) is a framework that runs an optimization process by connecting the optimization process to multiple CAD and CAE systems to change and evaluate the design. The design is modified, and analyzed under control of the PIDO.

Random

The term random is generally related to an event in which all outcomes are equally likely. A random number generator creates numbers that fill with a uniform distribution a specified range.

Reliability

Reliability is the probability of satisfying a requirement.

Residual

A residual is given by the difference between the fitted function (RSM) and the real points.

Response Surface Methodology

The Response Surface Methodology (RSM) is a collection of mathematical and statistical techniques useful for constructing approximation problems. Examples include low-order polynomials, gaussian processes, neural network. Once the approximations have been constructed, they are typically inexpensive to evaluate. RSM is used in engineering design to construct approximations of analysis codes in order to accelerate engineering design optimization. The approximations made within the observed space of variable values are called interpolations. On the contrary, approximation made outside the observed values are called extrapolations and require caution.

Robustness

Robustness is the ability to handle variation without loss of performance.

Robustness (algorithm)

The robustness of an optimization algorithm is the capability of correctly reach the absolute extreme of the objective functions from any imaginable starting condition.

Robust Optimization

In many real world optimization problems, the design parameters are not fixed and are normally identified by the mean and the standard deviation. The presence of uncertainty makes the traditional approaches to optimization insufficient. Robust optimization is a modeling methodology, combined with a suite of computational tools, which is aimed to solve problems where there are uncertainties involved in the data or in the model. The importance of controlling variability as opposed to just optimizing the expected value is well recognized since many real world optimization problems in numerous disciplines and application areas, contain uncertainty.

Robust Solution

A solution is consider robust (or stable) if it is not influenced by small variation of the decision variables.

Scatter Plot

A scatter plot is a 2-dimensional chart, it reveals relationships between two variables. It shows immediately if two variables are related, if the relation is linear or non-linear, if there are outliers or anomalies in the database.

Sequential Quadratic Programming

SQP is an optimization algorithm that finds a step away from the current point by minimizing a quadratic model of the problem.

Six Sigma

Six Sigma is a vision of quality which equates with only 3.4 defects per million opportunities for each product or service transaction. Six Sigma strives for perfection.

Skewness

The skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable.

Stable Solution

A solution is consider robust (or stable) if it is not influenced by small variation of the decision variables.

Standard Deviation

The Standard Deviation characterizes the dispersion among the measures in a given population. To calculate the standard deviation of a set of data it is first necessary to calculate the variance. Numerically, the standard deviation is the square root of the variance. Unlike the variance, which is a somewhat abstract measure of variability, the standard deviation can be readily conceptualized as a distance along the scale of measurement.

Stochastic

Involving a random variable.

Taguchi

Taguchi is an engineer and statistician. From the 1950s onwards, Taguchi developed a methodology for applying statistics in order to improve the quality of manufactured goods. He has made important and valuable methodological innovations in statistics and engineering especially with the introduction orthogonal arrays.

Taguchi Method

The Taguchi Method is a simple optimization method that uses a fixed number of evaluations and a summation technique to calculate an improved solution. It is designed to improve the quality of a product or process by not only striving to achieve performance targets, but also minimizing performance variation.

Uncertainty

Uncertainty is due to errors in measuring, or difficulties in sampling, or moreover can depend on events and effects in the future that cannot be known with certainty (uncontrollable disturbances and forecasting errors). In many engineering design problems, the design parameters may only be known to some tolerance or in some cases they may be described by a probability distribution. Moreover, designing a product for a specific environmental scenario does not guarantee good performance for other environments: there is a risk associated with the chosen design; another design may have a lower risk. Deterministic approaches to optimization do not consider the impact of such variations, and as a result, a design solution may be very sensitive to these variations. These uncertainties should be included in the optimization procedures, so that prescribed robustness can be achieved in the optimal designs.

Utopia Point

The utopia point is the theoretical optimum that is not achievable in objective space. It consists in the combination of the best instances of each objective function into a single point.

Variance

Variance is one of several indexes of variability used to characterize the dispersion among the measures in a given set. To calculate the variance of a given set, it is necessary to first calculate the mean, then to measure the amount that each value deviates from the mean, and then to square that deviation. Numerically, the variance equals the average of the several squared deviations from the mean.