MSE-415: B. Hawrylo Chapter 13 – Robust Design What is robust design/process/product?: A robust product (process) is one that performs as intended even under non-ideal conditions such as process variations or a range of operating situations. !

MSE-415: B. Hawrylo Chapter 13 – Robust Design What is robust design/process/product?: For a given performance target there may be many combinations that yield desired results. But, some combinations may be more sensitive to variation than others. Which combinations work the best? Which combinations have the most ‘robust’ Design to meet customer needs?

MSE-415: B. Hawrylo Chapter 13 – Robust Design The robust design process has 7 primary steps: 1.Identify Signal Factor(s), Response Variable or Ideal Function, Control Factors, Noise Factors, and Error States (or the failure modes). 2.Formulate an objective function. 3.Develop the experimental plan. 4.Run the experiment. 5.Conduct the analysis. 6.Select and confirm factor setpoints. 7.Reflect and repeat.

MSE-415: B. Hawrylo P-Diagram The P-Diagram is based on the concept of converting 100% of input energy (input signal) into 100% of the ideal function. Any engineered system reaches its "ideal function" when all of its applied energy (input) is transformed efficiently into creating desired output energy. In reality, nothing functions like this. Every system is less than 100% efficient in its energy transformation. This loss goes to creating unintended functions, or error states. Signal Ideal Function Noise Error States =

MSE-415: B. Hawrylo Chapter 13 – Robust Design Identify Signal Factor(s), Response Variable or Ideal Function, Control Factors, Noise Factors, and Error States (or the failure modes). 1.Signal Factor (inputs) pass through the design of the product and is output into measured Response Variable or Ideal Function. 1.The Signal Factor is transformed via the Control Factors to convert the input to the desired Output. 2.Control Factors are typically elements such as design, materials and processes that the engineer has 'control' over. 3.Error States are the Failure Modes or Effects of Failure as defined by an end user when using the product. 4.Noise Factors are things that can influence the design but are not under the control of the engineer, such as environmental factors, customer usage, interfaces with other systems, degradation over time, piece-to-piece variation, among others. These 1.Noise Factors, if not protected for, can make the design useless and it can be said that the design is not robust against the expected noise factors.

MSE-415: B. Hawrylo Example I want to determine how long it takes to boil enough water for a of cup tea. – Complete the Parameter Diagram

MSE-415: B. Hawrylo Chapter 13 – Robust Design 2. Formulate an objective function. An experiments performance metrics must be turned into an objective function that relates to the desired robust performance. Maximizing: Used for performance dimensions where larger values are better. Example? Minimizing: Used for performance dimensions where smaller values are better. Example? Target Value: Used for performance dimensions where values closest to a point are better. Example? Signal-to-noise ratio: Used to measure robustness. Signal Ideal Function Noise Error States = pg. 271

MSE-415: B. Hawrylo Chapter 13 – Robust Design 3. Develop the experimental plan.

MSE-415: B. Hawrylo Step 3: DOE DOE a method for obtaining the maximum amount of information for the least amount of data (saving resources, money and time) Basic question – What is the effect on Y when changing X? Simplest case: one factor, pick various levels of X (hold other X’s constant)

MSE-415: B. Hawrylo Which levels of X should be picked? First ask what is the range of interest? Statistical model valid only inside the experimental range (Can NOT extrapolate)

MSE-415: B. Hawrylo How many level of X to pick? What is happening between the two points?

MSE-415: B. Hawrylo How many level of X to pick? Possibilities include

MSE-415: B. Hawrylo Confounding Confounding is when a factorial experiment is run in blocks and the blocks are too small to contain a complete replication of the experiment, one can run a fraction of the replicate in each block, but the result is losing information on some effects. When two factors are varied such that their individual effects can not be determined separately, their effects are said to be confounding What if there are more than one independent variable? Which levels should be picked? Clearly there is a trend. But is the increase in Y due to an increase in X1 or an increase in X2 or both Note: In this situation, X1 and X2 are said to be confounding.

MSE-415: B. Hawrylo Orthogonal Design A better way to set-up the experiment is to use an orthogonal design. Clearly there is a trend. Increasing X1 causes Y to increase, while increasing X2 cause Y to decrease

MSE-415: B. Hawrylo What if there are many X’s? What if you had this design: X1 – 7 levels X2 – 5 levels X3 – 2 levels X4 – 2 levels This design would require 7x5x2x2 = 140 cases With more variables this can get out of hand. Fortunately DOE offers a better way. Examples? ???

MSE-415: B. Hawrylo DOE Process Early stage of DOE  Which factors are important and what is their overall effect? Later stages of DOE  What is the exact relationship between each factor and its response?  How can we optimize combinations of factor levels to maximize or minimize a response?

MSE-415: B. Hawrylo DOE Approach #1 All Possible Combinations Measure effect of one particular factor by fixing levels of remaining factors and running experiments at various levels of factors of interest. Repeat entire process for each of the other factors, one at a time. This allows one to measure the “simple effect” of X’s on Y. Problem: too many runs needed. In our case 140 runs

MSE-415: B. Hawrylo DOE approach #2 2 k Factorial Design Choose two levels for each of k possible factors and run experiments at each of the 2 k factor-level combinations Allows us to estimate the “main effect” on X’s on Y. Advantage: Less runs required.

MSE-415: B. Hawrylo 2k Factorial Design Two levels of factors are denoted by “-” or “low” level and “+” or “high” level, respectively Example: 3 factors (2 3 design) Note: R1….R8 are values of the response associated with the i’th combination of factor levels.

MSE-415: B. Hawrylo Main Effect (ME) Formal definition: average change in the response due to moving a factor from its “-” level to its “+” level while holding all other factors fixed For the previous example, ME of X1 can be calculated as follows: ME(X1) = (R2 – R1) + (R4 - R3) + (R6 - R5) + (R8 - R7) 4

MSE-415: B. Hawrylo Interaction Effect Sometimes two factors can interact with each other. Consider the following case X1 is varied over 4 levels, X2 is held constant – Y appears to be increasing as a function of X1. 1

MSE-415: B. Hawrylo Interaction Effect Suppose the experiment is repeated for a different level of X2: Y appears to be decreasing as a function of X1 (the exact opposite of the previous case)

MSE-415: B. Hawrylo Interaction Effect Interactions  Effect of one factor (X1) depends on level of another factor (X2)  Synergistic or antagonistic  Determine via plots or statistical test  Higher order interactions are possible but rare

MSE-415: B. Hawrylo Fractional Factorial Designs What if there are many factors? Clearly this can get out of hand!

MSE-415: B. Hawrylo Fractional Factorial Designs Is it necessary to run every single combination of every single factor at every single level? Fortunately, the answer is NO? Fractional Factorial Design allows us to get good estimates of main effects and interactions at a fraction of the price or time and money! A certain subset of the 2k possible design points are selected. But, which ones to choose? Theoretical statistics give the answer and computer programs do the work.

MSE-415: B. Hawrylo Fractional Factorial Designs How does it work?  Price of fractional factorial: certain effects confounded with each other  Example: – Main effect confounded with interaction:  Assumption: – Higher order interactions are negligible main effects and lower order interactions

MSE-415: B. Hawrylo Fractional Factorial Designs If we can assume that higher-order interactions are negligible, we don’t need to run every single case. Subject matter expert/analyst is consulted to determine which interactions are likely to be negligible. Design (run matrix) is chosen accordingly. DOE Strategy  Objective: screen out unimportant factors, identify significant ones  Another name for a fractional factorial design is “screen design”  Original design matrix allows us to identify significant factors – BUT-

MSE-415: B. Hawrylo DOE Strategy What is the exact nature of the relationship between these factors and the response? Original design has 8 design points

MSE-415: B. Hawrylo DOE Strategy Additional data points can be added to the original design A “follow-up” study can be done to optimize or determine sensitivity of a response. Other types of design exist.

MSE-415: B. Hawrylo Other Designs Mixture/Simplex Design  Levels of X’s add up to 1 (not independent)  Commonly used in chemistry industry  Could be used to optimize shipfill Computer-Generated (Optimal) Design  Irregular experimental design (e.g., region of interest is not a cube or a sphere due to constraints on X’s)  Nonstandard model  Unusual sample size requirement

MSE-415: B. Hawrylo Step 4: Conduct the experiment. Vary the control and noise factors Record the performance metrics Compute the objective function Step 4: Run the Experiment

MSE-415: B. Hawrylo Paper Airplane Experiment

MSE-415: B. Hawrylo Step 5: Perform analysis of means. Compute the mean value of the objective function for each factor setting. Identify which control factors reduce the effects of noise and which ones can be used to scale the response. (2-Step Optimization) Step 5: Conduct Analysis

MSE-415: B. Hawrylo Analysis of Means (ANOM) Plot the average effect of each factor level.  Choose the best levels of these factors Scaling factor? Prediction of response: E  (Ai, Bj, Ck, Dl)] =  + a i + b j + c k +d l

MSE-415: B. Hawrylo Step 6: Select control factor setpoints. Choose settings to maximize or minimize objective function. Consider variations carefully. (Use ANOM on variance to understand variation explicitly.) Step 6: Select Setpoints