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[交流] 【分享】Design of experiments

28 August 2008

Prof Jack P.C. Kleijnen
Tilburg University (UvT) / School of Economics & Business (FEB), Netherlands
http://center.uvt.nl/staff/kleijnen/

Design Of Experiments (DOE) is needed for experiments with

real-life (physical) systems in agriculture, chemical plants, industrial factories, service industries, etc. (Montgomery 2009);
deterministic simulation models of airplanes, automobiles, TVs, computer chips, etc.—in Computer Aided Engineering (CAE) and Computer Aided Design (CAD)—at companies like Boeing, General Motors, Philips, etc. (Santner, Williams, and Notz 2003, Kleijnen 2008);
random (stochastic, discrete-event) simulation models of queuing and inventory systems (Kleijnen 2008, Law 2007).
This Page focuses on experiments with simulation models (either deterministic or random). These experiments may vary hundreds of factors (inputs variables and parameters)—each with many values. Consequently, a multitude of scenarios (combinations of factor values) might be simulated, were it not that the simulation of a single scenario may require too much computer time. Simulation experiments are well-suited to sequential designs (discussed below) instead of “one shot” designs Classic designs are often fractional factorials; i.e., only a fraction of all conceivable scenarios are simulated. For example, if there are 7 factors and each factors is observed at 2 values only, then there are still 128 possible combinations; actually, only 8 combinations suffice to estimate the first-order or “main” effects of these factors. Notice that changing only 1 factor at a time also requires only 8 combinations, but it can be proven that a fractional factorial gives estimates that are less sensitive to experimental noise (usually assumed to be “white” noise; see below). These classic designs may also allow interaction among factors and second-order effects

Simulation designs also use Latin Hypercube Sampling (LHS), which allows more complicated Input/Output (I/O) behavior of the underlying simulation model. LHS provides a “space filling” design, whereas classic designs select extreme scenarios at the corners of the experimental space. Space filling designs are often analyzed through Kriging (spatial correlation or Gaussian) models. Kriging gives an exact interpolator; i.e., for “old” scenarios the Kriging prediction equals the observed outputs

Which design is “best” depends on the goal of the experiment: sensitivity analysis, optimization, etc. Sensitivity analysis may serve validation & verification of the underlying simulation model, and factor screening, which is the search for the really important factors. Optimization tries to find the “best” scenario for the simulated real system. Robust optimization allows for uncertain environmental factors.

DOE assumes white noise;i.e., normally and independently distributed noise with a constant variance. In practice, however, this assumption is often unrealistic; e.g., the experimental output (response) may have variability that is not constant when the input combination changes. The experimenters should therefore test whether the white-noise assumption holds; if the assumption does not hold, the experimenters may adapt their analysis methods. For example, Weighted Least Squares (WLS) weight the experimental outputs with their variability. Further analysis is possible through computer-intensive methods, such as “bootstrapping” and “cross-validation” (Kleijnen 2008).

Screening is related to “sparse” effects, “parsimony”, “Pareto’s principle”, “Occam’s razor”, “20-80 rule”, and “curse of dimensionality”. An example is provided by a specific greenhouse simulation model with 281 factors. The users (politicians) want to focus on a few really important factors. Even a classic design for first-order effects would require the simulation of at least 282 scenarios—which would take too much computer time.

Many screening designs are sequential (not one-shot); i.e., observations are analyzed before the next scenario is selected and observed. This approach implies that the design is customized, tailored, application-driven, or non-generic. For Kriging there are also sequential designs, which are indeed customized; e.g., the design selects relatively many scenarios with non-linear I/O behavior.

For optimization there are many methods (Kleijnen 2008). Unfortunately, most methods for optimization in simulation assume a single output, whereas in practice simulation models give multiple outputs. A recent variant of Response Surface Methodology (RSM) allows multiple outputs, and searches for the optimal input combination in a sequence of steps (using estimated local gradients). An alternative method combines Kriging and Mathematical Programming. This alternative uses (i) sequential designs to specify the simulation inputs, (ii) Kriging to analyze the global I/O behavior, and (iii) Integer Non-Linear Programming to estimate the optimal solution from the Kriging models.

Whereas most optimization methods assume known (fixed) environments, robust methods allow uncertain environmental factors. The Japanese engineer Taguchi first applied such a view; his statistical techniques have been improved later on by others (Kleijnen 2008, Myers and Montgomery 1995). Robust optimization may minimize the variability of the output subject to a restriction on the expected output, to select a compromise or "robust" solution.

In conclusion, simulation allows experimentation with many factors, so screening designs are very important. Kriging allows more complicated I/O behavior of the system. Sequential designs for either sensitivity analysis or optimization are more efficient than one-shot designs. Optimization should account for multiple outputs. Robust optimization is important in an uncertain world.

Design Of Experiments (DOE) is needed for experiments with

real-life (physical) systems in agriculture, chemical plants, industrial factories, service industries, etc. (Montgomery 2009);
deterministic simulation models of airplanes, automobiles, TVs, computer chips, etc.—in Computer Aided Engineering (CAE) and Computer Aided Design (CAD)—at companies like Boeing, General Motors, Philips, etc. (Santner, Williams, and Notz 2003, Kleijnen 2008);
random (stochastic, discrete-event) simulation models of queuing and inventory systems (Kleijnen 2008, Law 2007).
This Page focuses on experiments with simulation models (either deterministic or random). These experiments may vary hundreds of factors (inputs variables and parameters)—each with many values. Consequently, a multitude of scenarios (combinations of factor values) might be simulated, were it not that the simulation of a single scenario may require too much computer time. Simulation experiments are well-suited to sequential designs (discussed below) instead of “one shot” designs Classic designs are often fractional factorials; i.e., only a fraction of all conceivable scenarios are simulated. For example, if there are 7 factors and each factors is observed at 2 values only, then there are still 128 possible combinations; actually, only 8 combinations suffice to estimate the first-order or “main” effects of these factors. Notice that changing only 1 factor at a time also requires only 8 combinations, but it can be proven that a fractional factorial gives estimates that are less sensitive to experimental noise (usually assumed to be “white” noise; see below). These classic designs may also allow interaction among factors and second-order effects

Simulation designs also use Latin Hypercube Sampling (LHS), which allows more complicated Input/Output (I/O) behavior of the underlying simulation model. LHS provides a “space filling” design, whereas classic designs select extreme scenarios at the corners of the experimental space. Space filling designs are often analyzed through Kriging (spatial correlation or Gaussian) models. Kriging gives an exact interpolator; i.e., for “old” scenarios the Kriging prediction equals the observed outputs

Which design is “best” depends on the goal of the experiment: sensitivity analysis, optimization, etc. Sensitivity analysis may serve validation & verification of the underlying simulation model, and factor screening, which is the search for the really important factors. Optimization tries to find the “best” scenario for the simulated real system. Robust optimization allows for uncertain environmental factors.

DOE assumes white noise;i.e., normally and independently distributed noise with a constant variance. In practice, however, this assumption is often unrealistic; e.g., the experimental output (response) may have variability that is not constant when the input combination changes. The experimenters should therefore test whether the white-noise assumption holds; if the assumption does not hold, the experimenters may adapt their analysis methods. For example, Weighted Least Squares (WLS) weight the experimental outputs with their variability. Further analysis is possible through computer-intensive methods, such as “bootstrapping” and “cross-validation” (Kleijnen 2008).

Screening is related to “sparse” effects, “parsimony”, “Pareto’s principle”, “Occam’s razor”, “20-80 rule”, and “curse of dimensionality”. An example is provided by a specific greenhouse simulation model with 281 factors. The users (politicians) want to focus on a few really important factors. Even a classic design for first-order effects would require the simulation of at least 282 scenarios—which would take too much computer time.

Many screening designs are sequential (not one-shot); i.e., observations are analyzed before the next scenario is selected and observed. This approach implies that the design is customized, tailored, application-driven, or non-generic. For Kriging there are also sequential designs, which are indeed customized; e.g., the design selects relatively many scenarios with non-linear I/O behavior.

For optimization there are many methods (Kleijnen 2008). Unfortunately, most methods for optimization in simulation assume a single output, whereas in practice simulation models give multiple outputs. A recent variant of Response Surface Methodology (RSM) allows multiple outputs, and searches for the optimal input combination in a sequence of steps (using estimated local gradients). An alternative method combines Kriging and Mathematical Programming. This alternative uses (i) sequential designs to specify the simulation inputs, (ii) Kriging to analyze the global I/O behavior, and (iii) Integer Non-Linear Programming to estimate the optimal solution from the Kriging models.

Whereas most optimization methods assume known (fixed) environments, robust methods allow uncertain environmental factors. The Japanese engineer Taguchi first applied such a view; his statistical techniques have been improved later on by others (Kleijnen 2008, Myers and Montgomery 1995). Robust optimization may minimize the variability of the output subject to a restriction on the expected output, to select a compromise or "robust" solution.

In conclusion, simulation allows experimentation with many factors, so screening designs are very important. Kriging allows more complicated I/O behavior of the system. Sequential designs for either sensitivity analysis or optimization are more efficient than one-shot designs. Optimization should account for multiple outputs. Robust optimization is important in an uncertain world.

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