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SYSEN 5300
Assignment 8 / Takehome Final
Dynamic Fault Tree, Factorial Design at Two Levels and
Response Surface Method
Instructions:
This assignment is weighted as the final exam.
You can work in teams, as with the other assignments. Only one submission is required from
each team. Remember to include all your works for the full credits.
You should seek help from no source other than
• Lecture notes and readings posted on Blackboard,
• Texts suggested in the syllabus,
• Your other assignments from SYSEN 5300,
• JMP or a data analysis software of your choice, unless otherwise indicated.
• The documentation of JMP or your data analysis software,
• TA or Instructor (on Piazza—we will not be responding to email).
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Question 1: Factorial design at two levels. (30 points)
An experiment was conducted on an operating chemical process in which four factors were studied
in a 24 factorial design. Shown in the table below are the factor levels, the design, the random order
in which the runs were made, and the response (impurity) at each of the 16 reaction conditions:
Factor Levels - +
1 Concentration of Catalyst (%) 5 7
2 Concentration of NaOH(%) 40 45
3 Agitation speed (rpm) 10 20
4 Temperature (F) 150 180
Coded Factor Levels of the Experiment Design
Run Order 1 2 3 4 Impurity
2 - - - - 0.38
6 + - - - 0.4
12 - + - - 0.27
4 + + - - 0.3
1 - - + - 0.58
7 + - + - 0.56
14 - + + - 0.3
3 + + + - 0.32
8 - - - + 0.59
10 + - - + 0.62
15 - + - + 0.53
11 + + - + 0.5
16 - - + + 0.79
9 + - + + 0.75
5 - + + + 0.53
13 + + + + 0.54
(a) Make the table of contrast (a table of plus and minus signs from which all the main
effects and interaction effects be calculated).
(b) Calculate all the main and interaction effects. Show the steps of your calculation and
only run software to check your answer.
(c) Assuming the three-factor and higher order interactions to be noise, compute an estimate
of the error variance of the effects. Show the steps of your calculation and only run
software to check your answer.
(d) Based on your results from Question 1(b) and (c), which of the estimated effects are
likely to be the real effects rather than noise? Why? How would you interpret each of the
real effect(s)? Show your interpretation graphically.
Hint: see slide 14-15 or reading material Section 5.8.
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(e) Assume that the present conditions of operation are 1=5%, 2=40%, 3=10 rpm, 4=180F.
Make recommendations of better operation conditions to reduce the impurity.
(f) Assume your conclusions from Question 1(d) remain valid for operation conditions
beyond the range enclosed by the two specified levels of the four factors. If you are able
to conduct a few more experiment runs, describe how you would set and/or vary the four
factors in the additional experiments so that, with as few experiments as possible,
i. You may confirm/ further investigate each of the real effects found in Question
1(d), and
ii. You could most likely find even better operation conditions than your answer to
Question 1(e).
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Question 2: Application of response surface method. (30 points)
The data table below came from a tire radial run-out study. Lower run-out values are desirable.
The factors under study were:
• = Post-inflation time (minutes),
• = Post-inflation pressure (psi),
• = Cure temperature (0F),
• = Rim size (inches), and
The response variable was:
• = Radial run-out (mils).
1 9.6 40 305 4.5 0.51
2 32.5 40 305 4.5 0.28
3 9.6 70 305 4.5 0.65
4 32.5 70 305 4.5 0.51
5 9.6 40 335 4.5 0.24
6 32.5 40 335 4.5 0.38
7 9.6 70 335 4.5 0.45
8 32.5 70 335 4.5 0.49
9 9.6 40 305 7.5 0.3
10 32.5 40 305 7.5 0.35
11 9.6 70 305 7.5 0.45
12 32.5 70 305 7.5 0.82
13 9.6 40 335 7.5 0.24
14 32.5 40 335 7.5 0.54
15 9.6 70 335 7.5 0.35
16 32.5 70 335 7.5 0.51
17 60 55 320 6 0.53
18 5 55 320 6 0.56
19 17.5 85 320 6 0.67
20 17.5 25 320 6 0.45
21 17.5 25 350 6 0.41
22 17.5 25 290 6 0.23
23 17.5 25 320 9 0.41
24 17.5 25 320 3 0.47
25 17.5 25 320 6 0.32
1x
2x
3x
4x
y
1x 2x 3x 4x y
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(a) Suppose you want to fit a second-order polynomial model to the data. Write the
equations for least square regression in vector/matrix form. Define all the variables in
your equation and specify the dimensions of each. Write down the matrix of independent
variable values, .
Hint: a second-order polynomial model can be written as
.
(b) Make what you think is an appropriate analysis of the data to obtain your best model for
radial run-out, :
i. Describe your approach and steps of analysis.
ii. Report your model expression, the overall goodness-of-fit (e.g. ANOVA, esp.
Rsquare and Adjusted Rsquare), estimated parameters, and their significance
results (t-ratio or p-value).
Hint: use stepwise least square regression. The terms included in the final model should
be either a significant variable or part of a significant variable.
(c) Based on your final model, make a two-dimensional contour plot of versus and
over the ranges and , for each of the following three sets of
conditions:
i. = 40, = 3
ii. = 55, = 9
iii. = 55, = 4.5
Condition iii is the current condition used in the plant for production.
(d) Comment on this conclusion: "It was surprising to observe that either very wide (9-inch)
or very narrow (3-inch) rims could be used to reach low radial run-out levels". Is it true?
Use you model to explain why it is true/not true.
(e) Based on your model, how you would vary the factors studied above to achieve low
radial run-out? Besides the model results, discuss the practical/operational
considerations in determining how the factors should be varied.
X
2 2
0 1 1 2 2 3 3 4 4 5 1 8 4 9 1 2 14 3 4ˆ ... ...y x x x x x x x x x xb b b b b b b b b= + + + + + + + + + +
y
yˆ 1x 3x
605 1 ££ x 3290 350x£ £
2x 4x
2x 4x
2x 4x
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Question 3: More than one response variables. (20 points)
The coded factor settings and results from an experiment are shown below. The objective is to look
for the settings to achieve a high yield and low filtration time.
Trial Yield (gs)
Field
Color
Crystal
Growth
Filtration
Time
(sec) 1 - - 21.1 Blue None 150
2 - 0 23.7 Blue None 10
3 - + 20.7 Red None 8
4 0 - 21.1 Slightly red None 35
5 0 0 24.1 Blue Very slight 8
6 0 + 22.2 Unobserved Slight 7
7 + - 18.4 Slightly red Slight 18
8 + 0 23.4 Red Much 8
9 + + 21.9 Very red Much 10
Variable Level
Variable - 0 +
= Condensation temperature (C) 90 100 110
= Amount of B (cm3) 24.4 29.3 34.2
(a) Analyze the data to obtain your best model:
i. Describe your approach and steps of analysis.
ii. Report your model expression, the overall goodness-of-fit (e.g. ANOVA, esp.
Rsquare and Adjusted Rsquare), estimated parameters, and their significance results
(t-ratio or p-value).
Hint: fit a second-order polynomial model by stepwise least square regression with each of
the two responses. Take the logarithm of filtration time to fit a better model.
(b) Find the settings of and within the range enclosed by the experiment design levels that
give
i. The highest predicted yield, and
ii. The lowest predicted filtration time.
Hint: optimize each response separately. If you use an analytical method, you should check
whether the true maximum/minimum condition is satisfied.
1x 2x
1x
2x
1x 2x
Question 4: Dynamic Fault Tree (20 points)
A farm uses drones to monitor crop health and apply pesticides as needed. The farm is divided
into two plots, one north and one south. The north section is larger and has two drones assigned
to it, whereas the south section has only one drone assigned to it. Additionally, there are two
drones held in reserve, which act as pooled spares which replace failed drones on either plot.
(a) Assume that all drones have a constant failure rate of = 10!" per hour. A Markov chain for
the farm is shown below. The numbers represent the number of drones assigned to north, south,
and in reserve, respectively. Fill in the transition rates between states. Calculate the expected time
of mission failure, and the probability that the north plot fails first.
(b) Suppose the drones acting as pooled spares have a failure time described by a lognormal
distribution with a MTTF of 10" hours. Is the expected time of mission failure the same? Explain
why or why not.
(c) Simulate the time of mission failure for the situation described in part B, where the pooled
spares have a lognormal distribution with = ln(1000) − 2.25/2 ≈ 5.7828, = 1.5 and the
other 3 drones have a constant failure rate of = 10!". Give the expected time of mission failure
averaged over 10,000 simulations.
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Hint: The mission of protecting the crops is considered failed if either the north or
south plot has 0 drones assigned to it. Note that while a drone is being held as a pooled
spare (meaning it is not actively deployed) there is no risk of that drone failing.
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Assignment 8 / Takehome Final JMP Tips
Analyzing experiments of two-level full factorial design using least square regression in JMP:
1. Set up the coded factor levels from DOE>Custom Design or DOE>Full Factorial Design
and make the design table.
2. Enter the response values from the experiments the design table.
3. Run the script "Model" from the generated design table or click Analyze>Fit model.
4. Check and configure the model specification in the pop-up window and click "Run".
5. Note that an effect is half the value of the corresponding parameter estimated in least square
regression.
6. In the model results window, red triangle next to "Prediction Profiler" > Interaction Profiler
will make the interaction profiles appear.
To take the logarithm of the original response data as the response values for fitting the model:
Model Specification>Select response variable>Click “Y” under “Pick Role Variables”>Click the
response variable that appears next to “Y”>Click the red triangle next to “Transform” at the bottom
of the window, under “Construct Model Effects”>Click “Log”.
This way, you can use the logarithm of the original data to fit the model without adding a Log(data)
column to your data table, and the output prediction expression is also transformed back.