🧰Engineering Applications of Statistics Unit 8 – Design of Experiments in Engineering Stats

Key Concepts and Terminology

• Design of Experiments (DOE) involves planning, conducting, and analyzing experiments to optimize processes or products
• Factors are the independent variables manipulated in an experiment (temperature, pressure)
• Levels are the specific values or settings of a factor (low, medium, high)
• Response variable is the dependent variable measured to evaluate the effect of factors (yield, strength)
• Main effect is the individual effect of a factor on the response variable
• Interaction effect occurs when the effect of one factor depends on the level of another factor
• Replication involves repeating the entire experiment or individual runs to estimate experimental error
• Randomization is the random assignment of experimental units to treatments to minimize bias
• Blocking is a technique used to reduce the impact of nuisance factors on the response variable

Experimental Design Principles

• Randomization ensures that the allocation of experimental units to treatments is unbiased
  • Helps to minimize the effect of unknown or uncontrollable factors
• Replication allows for the estimation of experimental error and increases the precision of the results
  • Provides a measure of the inherent variability in the process or system
• Blocking is used to reduce the variability caused by nuisance factors (batch, operator)
  • Groups similar experimental units together to minimize the impact of these factors
• Factorial designs enable the study of multiple factors simultaneously and their interactions
• Confounding is a design technique that deliberately confuses the effects of certain factors with blocks
  • Allows for the efficient use of resources while maintaining the ability to estimate main effects
• Orthogonality ensures that the effects of different factors can be estimated independently
• Balanced designs have an equal number of observations for each treatment combination

Types of Experimental Designs

• Completely Randomized Design (CRD) is the simplest design, where treatments are randomly assigned to experimental units
• Randomized Complete Block Design (RCBD) uses blocking to reduce the impact of nuisance factors
  • Each block contains a complete set of treatments
• Latin Square Design (LSD) is used when there are two nuisance factors (row and column effects)
  • Each treatment appears exactly once in each row and column
• Split-Plot Design is used when some factors are harder to change than others (whole-plot and sub-plot factors)
• Nested Design is used when the levels of one factor are different for each level of another factor
• Fractional Factorial Design is a subset of a full factorial design that allows for the estimation of main effects and some interactions
  • Useful when the number of factors is large and resources are limited
• Plackett-Burman Design is a screening design used to identify the most important factors from a large set of potential factors

Factorial Designs and Their Applications

• Factorial designs allow for the simultaneous study of multiple factors and their interactions
• Full factorial designs include all possible combinations of factor levels (2^k for k factors at two levels)
• Fractional factorial designs are a subset of full factorial designs that maintain the ability to estimate main effects and some interactions
  • Useful when the number of factors is large and resources are limited
• Factorial designs are widely used in engineering to optimize processes or products
  • Helps to identify the most important factors and their optimal settings
• Interaction plots are used to visualize the presence and nature of interactions between factors
• Confounding is a technique used in fractional factorial designs to deliberately confuse the effects of certain interactions with blocks
  • Allows for the efficient use of resources while maintaining the ability to estimate main effects

Analysis of Variance (ANOVA) in Experiments

• ANOVA is a statistical method used to analyze the results of experiments and determine the significance of factors
• ANOVA partitions the total variability in the response variable into components attributable to different sources (factors, interactions, error)
• The F-test is used to compare the variance between treatments to the variance within treatments
  • A large F-value indicates that the treatment means are significantly different
• The p-value is the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true
  • A small p-value (typically < 0.05) suggests that the null hypothesis should be rejected and the factor is significant
• Main effects and interaction effects can be tested for significance using ANOVA
• Residual analysis is used to check the assumptions of ANOVA (normality, homogeneity of variance, independence)
• Multiple comparison procedures (Tukey's HSD, Dunnett's test) are used to determine which treatment means are significantly different from each other

Randomization and Blocking Techniques

• Randomization is the process of randomly assigning experimental units to treatments
  • Helps to minimize the effect of unknown or uncontrollable factors on the response variable
• Blocking is a technique used to reduce the impact of nuisance factors on the response variable
  • Groups similar experimental units together to minimize the variability within blocks
• Randomized Complete Block Design (RCBD) is a common blocking design
  • Each block contains a complete set of treatments, and treatments are randomly assigned within each block
• Latin Square Design (LSD) is used when there are two nuisance factors (row and column effects)
  • Each treatment appears exactly once in each row and column
• Incomplete Block Designs are used when the number of treatments is large and it is not feasible to include all treatments in each block
  • Balanced Incomplete Block Design (BIBD) ensures that each treatment appears an equal number of times and each pair of treatments appears together an equal number of times
• Blocking can increase the precision of the experiment by reducing the experimental error
• The efficiency of blocking can be measured by the relative efficiency (RE) compared to a completely randomized design

Optimization and Response Surface Methods

• Response Surface Methodology (RSM) is a collection of statistical and mathematical techniques used to optimize processes or products
• RSM involves designing experiments to fit a model that relates the response variable to the factors
  • The model is then used to find the optimal settings of the factors that maximize or minimize the response variable
• Central Composite Design (CCD) is a common RSM design that allows for the estimation of quadratic models
  • Consists of a factorial design, center points, and axial points
• Box-Behnken Design (BBD) is another RSM design that requires fewer runs than CCD
  • Useful when the factors are limited to three levels and the experimental region is irregular
• Steepest Ascent (or Descent) Method is used to sequentially move towards the optimum by following the path of steepest ascent (or descent) in the response surface
• Contour Plots and Surface Plots are used to visualize the response surface and identify the optimal region
• Desirability Functions are used to simultaneously optimize multiple response variables by combining them into a single desirability score

Real-World Applications in Engineering

• Design of Experiments is widely used in various engineering fields to optimize processes, products, and systems
• In manufacturing, DOE is used to optimize process parameters (temperature, pressure, feed rate) to improve product quality and reduce costs
  • Example: Optimizing injection molding parameters to minimize defects and cycle time
• In chemical engineering, DOE is used to optimize reaction conditions (catalyst, temperature, pressure) to maximize yield and selectivity
  • Example: Optimizing the synthesis of a pharmaceutical compound to improve yield and purity
• In aerospace engineering, DOE is used to optimize aircraft design parameters (wing shape, engine placement) to improve performance and fuel efficiency
  • Example: Optimizing the design of a turbine blade to maximize power output and durability
• In materials science, DOE is used to optimize the composition and processing of materials to achieve desired properties (strength, toughness, conductivity)
  • Example: Optimizing the heat treatment process of a metal alloy to maximize hardness and wear resistance
• In environmental engineering, DOE is used to optimize the design and operation of treatment systems (wastewater, air pollution) to minimize environmental impact and cost
  • Example: Optimizing the dosage of coagulants in a water treatment plant to improve removal efficiency and reduce sludge production
• DOE is also used in combination with other techniques such as simulation, machine learning, and optimization algorithms to solve complex engineering problems
  • Example: Using DOE to generate training data for a machine learning model that predicts the performance of a solar panel based on its design parameters