MANE 6313¶
Week 15, Module B¶
Student Learning Outcome¶
- Select an appropriate experimental design with one or more factors,
- Select an appropriate model with one or more factors,
- Evaluate statistical analyses of experimental designs,
- Assess the model adequacy of any experimental design, and
- Interpret model results.
Module Learning Outcome¶
Employ method of steepest descent (MSD) to locate the region of a minimum
Method of Steepest Descent Demonstration¶
- Provide demonstration of method of steepest descent
- Similar to Example 11.1 from textbook but simplified
- Only dealing with natural (uncoded variables) and ignoring coded variables
- Model description
- Second order model with 3 inputs and one output
- Range for \(x_1\) is 100 - 200
- Range for \(x_2\) is 0.5 - 0.75
- Range for \(x_3\) is 25 - 75
- The true model will only be used to generate response variable values and not used in the method of steepest descent
- The true model was created to contain a minimum with the range of the input variables
Steps in Steepest Descent Analysis¶
- First Order DOE
- Selection of Minimum Design Point
- Construction of Gradient Vector
- Gradient Vector at Minimum Point
- Selection of Step Size
- Construction of Search Path
- Implementation and Results of MSD
First Order DOE¶
- A single replicate of a \(2^3\) full factorial was conducted around the point (110, 1.0, 30).
- The data for building the model is
| x1 | x2 | x3 | y |
|---|---|---|---|
| 100 | 0.5 | 25 | 94.575 |
| 100 | 0.5 | 35 | 82.825 |
| 100 | 1.5 | 25 | 85.625 |
| 100 | 1.5 | 35 | 74.375 |
| 120 | 0.5 | 25 | 63.875 |
| 120 | 0.5 | 35 | 51.325 |
| 120 | 1.5 | 25 | 55.125 |
| 120 | 1.5 | 35 | 43.075 |
- The fitted linear model is
\[
\hat{y}=273.05-1.44x_1-11.20x_2-0.8x_3+0.01x_1x_2-0.004x_1x_3+0.05x_2x_3
\]
Minimum Point¶
- Select a starting point for the steepest method search
- Since minimum is desired, select design point with smallest value for y
(120, 1.5, 35)
Gradient Vector¶
- The gradient vector is found by taking partial derivatives with respect to each variable
\[
\begin{aligned}
\frac{d\hat{y}}{dx_1}&= -1.44+0.01x_2-0.004x_3\\
\frac{d\hat{y}}{dx_2}&= -11.20+0.01x_1+0.05x_3\\
\frac{d\hat{y}}{dx_3}&=-0.80-0.004x_1+0.05x_2
\end{aligned}
\]
- Notice that the gradient vector is a function of \(x_1\), \(x_2\), and \(x_3\).
Gradient Vector at DSM Starting Point¶
- Recall from the first order design of experiments, (120, 1.5, 35) had the smallest response value and will be starting for the deepest descent method
- The gradient vector at the starting point is
\[
\begin{aligned}
\frac{d\hat{y}}{dx_1}&= -1.44+0.01(1.5)-0.004(35)=-1.565\\
\frac{d\hat{y}}{dx_2}&= -11.20+0.01(120)+0.05(35)=-8.25\\
\frac{d\hat{y}}{dx_3}&=-0.80-0.004(120)+0.05(1.5)=-1.205
\end{aligned}
\]
-
The gradient defines the direction of steepest ascent; for the steepest descent, move in the opposite direction
-
Therefore, the search direction is \((1.565, 8.25, 1.204)\)
Step Size¶
- Selection of the step size is very subjective
- Montgomery recommends selecting the largest gradient component and fixing the step size at a convenient number
\[
\Delta x_2=0.5
\]
- The other step sizes are scaled proportionally to \(x_2\) for this example
\[
\begin{aligned}
\Delta x_1&= \Delta x_2\left(\frac{-\frac{d\hat{y}}{dx_1}}{-\frac{d\hat{y}}{dx_2}}\right)=0.5\left(\frac{1.565}{8.25}\right)=0.09485\\
\Delta x_3&= \Delta x_2\left(\frac{-\frac{d\hat{y}}{dx_3}}{-\frac{d\hat{y}}{dx_2}}\right)=0.5\left(\frac{1.204}{8.25}\right)=0.07297\\\end{aligned}
\]
Search Path¶
- Combining the starting point and step size, the search path is
\[
\begin{aligned}
x_1&=120+0.09485t\\
x_2&=1.5+0.5t\\
x_3&=35+0.07297t
\end{aligned}
\]
Search Results¶
| t | x1 | x2 | x3 | y |
|---|---|---|---|---|
| 0 | 120 | 1.5 | 35 | 43.075 |
| 1 | 120.09485 | 2 | 35.7297 | 39.11371494 |
| 2 | 120.1897 | 2.5 | 36.4594 | 35.82161728 |
| 3 | 120.28455 | 3 | 37.1891 | 33.198707 |
| 4 | 120.3794 | 3.5 | 37.9188 | 31.24498412 |
| 5 | 120.47425 | 4 | 38.6485 | 29.96044862 |
| 6 | 120.5691 | 4.5 | 39.3782 | 29.34510052 |
| 7 | 120.66395 | 5 | 40.1079 | 29.3989398 |
| 8 | 120.7588 | 5.5 | 40.8376 | 30.12196648 |
-
The minimum was found at step 6 with values (120.57,4.5,39.38)
-
This value should be in the neighborhood of the minimum
- A full second-order RSM design should be build centered at (120.57,4.5,39.38)