Design an eight-run fractional factorial design to

Question

Part A - Questions

Q1. In the following experiment, there are two factors, A and B. The response variable is carbon monoxide (CO) emission in an automobile. The design is shown below. What kind of design is employed here?                                                                        

(a) Full Factorial Design

(b) Central Composite Design

(c) Fractional Factorial Design

(d) Full Factorial Design at three levels  

Run	Factor		Response (y)
	A	B
1	-1	-1	54
2	-1	1	45
3	1	-1	32
4	1	1	47
5	-1	0	53
6	1	0	47
7	0	-1	51
8	0	1	41
9	0	0	39
10	0	0	44
11	0	0	42
12	0	0	40

Q2.  Using the design shown for Question 1, which of the following number pairs are correct?

(a) = 4 and = 4   

(b) = 12 and = 4 = 8

(c) = 4 and = 8   

(d) and = 4

Q3. In an engineering system optimization problem, people want to minimize the system response. However, all the input settings they may choose are subject to disturbance and uncertainty, and will therefore not be held precise and constant. Considering the presence of disturbances, it is better to look for a robust optimal setting, while keeping the minimization objective. Assume the disturbance level the same for all the input settings. If the response surface looks like below, which setting is the robust optimal setting:

(a) A;

(b) B;

(c) C;

(d) D.

Q4. We have a response surface as shown below. Which one of the following equations CANNOT possibly describe this surface?

Q5. Given the following design matrix, the y vector used in a least squares estimation is arranged as = [49,51,47,48, 48,52, 49, 46].               

Run	A	B		y
			I		II
1	-	-	49		48
2	-	+	51		52
3	+	-	47		49
4	+	+	48		46

We want to estimate all the main effects as well as the two factor interaction. So the appropriate matrix is

Q6. Recall the Epitaxial Layer Growth Experiment has four factors, namely = 4. The first factor is a qualitative factor, called susceptor-rotation method, which can only take two options (continuous or oscillating). If we now want to do a central composite design for this problem, which of the following choices of (the position of the star points) is most reasonable? Here, assume that a qualitative factor can take values at the center point as well.

(a) 1

(b) 1.414

(c) 1.732

(d) 2

Q7. Consider the following example, which is an unreplicated 2² design plus 5 center points. The variance for the pure curvature effect is

(a) 0.0027

(b) 0.022

(c) 0.025

(d) 0.09

Q8. Comparing quality improvement using quadratic loss function and the go/no-go approach, which one of the statements is most reasonable?

(a) Within the tolerance range, using the quadratic loss function or the no/no-go approach leads to the same quality improvement outcomes.

(b) Outside the tolerance range, using the quadratic loss function or the no/no-go approach leads to the same quality improvement outcomes.

(c) Using the go/no-go approach clearly differentiates the good products (go) from the bad products (no-go), and thus, it is preferred for quality improvement practice.

(d) Using the quadratic loss function leads to continuous improvement toward the target (nominal), and thus, it is preferred for quality improvement practice.

Q9. In a robust design, people consider the control-by-noise interactions important, because

(a) If the significant control-by-noise interactions are clear, it guarantees that all main effects are clear;

(b) The existence of significant control-by-noise interactions is the vital apparatus to shield away the influence of the noise effects;

(c) Estimating the control-by-noise interactions leads to a more cost-effective design;

(d) The control-by-noise interactions are more important under the larger-the-better or smaller-the-better problems, which are more prevalent in robust designs.

PART B: Problems - Clearly explain the solution to each problem.

Problem 1- [Note: This problem has three parts] Design an eight-run fractional factorial design to experiment to study the effect of the following four factors on yield: temperature (150 or 190°F), concentration (20% or 30%), catalyst (type-I or type-II), and stirring rate (50 or 90 rpm).

a) Write down the design matrix;

b) Write down a planning matrix.

c) What is the resolution of this design? Write down the effect aliasing relationship. Which effect(s) is clear?

Problem 2 - Given the following factor-and-level table

Symbol	Low level (-)	High level (+)
A	8	12
B	2.25	3.00

and the first order model as yˆ= 60 + 1.5x_A - 0.8x_B. Finish the following table and also determine at which point you should switch to a second order design.

Design points	Coded Variable		Natural Variable		Predicted	Observed
					yˆ	y
	A	B	ξA	ξB
Base (staring						56.1
point)	0.8
Step size (?)						59.5
Base +?
Base +2?						61.4
Base +3?						63.8
Base +4?						58.7

Problem 3 - Suppose that we have a second order model as

Conduct a canonical analysis and find the optimal response. Clearly show each step of your calculations.