This document discusses response surface methodology (RSM) and how it can be used to optimize a response with multiple factors. RSM combines different design of experiments (DOE) techniques to map the relationship between factors and responses. It involves performing an initial experiment, following the slope of steepest ascent to find an optimal region, and then using a central composite design within that region to estimate quadratic effects and locate the true optimum. The document provides a chemical engineering example to illustrate how RSM would be applied step-by-step to maximize chemical yield based on reaction time and temperature.

1. Lecture 19
Chapter 11: Response Surface
Methods

2. Part 1

3. Real DOE: Combination of Methods
• Rare that a DOE will consist of a single design, and one ANOVA table
giving a useful answer.
• A good DOE will produce as more questions as well as answers.
• More likely:
• Screening Experiment to reduce large list of factors to few
• Experiments to identify optimal settings from important factors
• Repeat these experiments as much as possible to further optimize
• Confirmation Experiments to validate optimal settings.

4. Response Surface Methods
• Response Surface Methodology (RSM) is a
great tool that combines together various
methods learned in course so far.
• Gives the researcher the ability to read
complicated relationships between many
potential factors (X’s) and predict an
optimal response (Y)
• Could even predict multiple responses (Y)!

5. Response Surface Methods Steps
1. Evaluate current or known region (operating
conditions)
2. Find and follow slope of Steepest Ascent for
optimum
3. Explore region of optimum for optimal
response
4. Repeat if best region is further still
Cannot be done as single experiment. Must be a
sequence with multiple experiments getting closer
to ideal.
Picture it: on a mountain trying to
find the peak!!

6. Finding the top of a mountain
• Imagine: we are on a large
mountain and have to reach the
summit.
• No rocky cliffs or trees or
dangerous animals obstructing us
• Extremely foggy, so can only see
about 10 meters around us at a
time.
• Best bet to find the top: start
climbing in the steepest direction
possible.
• Keep doing this will reach the top
eventually

7. Optimal Response = Mountain Summit
• Similar concept applies to finding optimal
response with complicated system
• Get bearings on current location, and identify a
slope
• Follow that slope until hitting some sort of a
peak
• Get bearings on that location to see if summit is
nearby.
• Keep climbing on another slope if necessary.
• Note: This could also be applied to finding a
minimum response or even to coming as close
as possible to a set target

8. Center Points
• A standard 2k experiment is useful for estimating Main Effects and
Interactions.
• Cannot estimate Quadratic effects
• Cannot easily estimate responses in between the 2 chosen factor
levels.
• Center Points added to a 2k experiment will allow for quadratic
estimates.
• Will allow ability to estimate inside of just the 2 factors.
• Will allow for estimate of our old friend, “naturally occurring
variation” (process error)

9. Center Points
• Add on to experiment multiple experimental runs where
a=b=c=…= 0

10. Part 2

11. Chemical Engineering Example
• A chemical process has two factors that can easily be controlled
(reaction time and reaction temperature)
• Current Reaction Time is 35 minutes (X1)
• Current Reaction Temperature is 155 degrees F (X2)
• Various factors exist that could be improved, but Yield is most
important right now. Currently around 40.
• Need to find the maximum possible Yield by altering Time and Temp.

12. Initial Experiment: Chem Engineering
Example
• Run 5 center points at current setting (35, 155)
• Corner points at +/5 time and +/- 5 temp
• (150, 160) and (30, 40)
• Can estimate Temp effect, Time Effect,
Interaction, and overall Quadratic (not
individual quadratics)
30
160
150
40
35, 155

13. Initial Experiment

14. First Experiment ANOVA (Regression)
• Main Time and Main Temp Effects
significant!
• Overall model has significance, no
real interaction or quadratic effect.
• Could be a slope happening here,
but no minimum or maximum.
• Note: This is set up in Minitab as a
simple 2 Factor Factorial design with
5 center points. Do not use
“Response Surface” platform … yet.

15. First Experiment: Contour
• Graphically, we appear to be
in the middle of a slope.
• Like on the mountain, can
approach Summit if we climb
up!
• This slope is called slope of
Steepest Ascent.

16. Part 3

17. Steepest Ascent: Which “Slope” is Steepest?
• Should take experimental
samples along “slope” within
Temp/Time space, as Yield will
increase.
• Analyze simple model on X1,
X2 (-1, 1) with no interactions
or quadratics.

18. Determining “Slope of Steepest Ascent”
1. Determine overall slope
within “X” space (coded
values)
2. Convert into actual units
3. Run experimental runs
moving upward along slope

19. Step 1: Determine Slope (Coded)
• Use regression coefficients to
determine slope.
• Slope of best line can be
found by dividing coefficients
• Slope=0.325/0.775=0.42
• Steepest Ascent will start at
(0,0) and go up 0.42 (X1) for
every 1 X2)

20. Slope of Steepest Ascent (or Descent)
• General formula (use if more than 2 X factors)
• Coded slope compared to other terms
∆𝑥𝑖 =
𝛽𝑖
𝛽𝑗
∆𝑥𝑗
=
0.325
0.775
1
= 0.42

21. Step 2: Convert to Actual Units
• Reminder: 1 unit of X1 (Time) = 5 minutes
• Reminder: 1 unit of X2 (Temperature) = 5 degrees
• Since slope=∆𝑥1 = 0.42
For every 5 minute increase in Time, Temp increase is:
5*0.42 ~= 2 degrees

22. 3. Following the Slope
80%!!

23. 3. Following the Slope
• Like climbing straight up a mountain
• Go up until you hit a peak.
• The actual highest point on the mountain
may be there, or it may be near there.

24. Is This the Optimal Value?
• Do we believe this is the best we can do?
• (Yield = 80.3, Time = 85, Temp = 175)
• Can further optimize in this area.

25. Peak, or maybe Relative Peak
Relative Peak may
be here
Need to find
Summit

26. Part 4

27. Benefit of Following Slope
• Initial Experiment
identifies a slope for
improvement
• Following slope
identifies potential
optimal region.
• Massive experiments
over entire
Time/Temp space:
inefficient!
30 50 70 90 110
150
160
180
170
Previously Known Region
Potentially High Yield Region

28. Next Step: Further Optimization?
• Simple Factorial with center points
again.
• NOTE: X1 and X2 have different
meanings now.
• Centered around (Time, Temp) =
(85, 175)
• Are we on a slope? Are we near a
peak?
• A peak would be present if there is a
quadratic effect.

29. Factorial Results
• Quadratic and Interaction are
significant.
• Sign could be close to a “peak” of
desired Yield variable.
• Cannot really estimate Optimal
values, since we only have estimate of
overall quadratic, not each individual
X.

30. Central Composite Design
• Need to estimate Main
Effects, Interactions, and
Quadratic Effects
(curvature) for each Factor
• Can add to the previous
Factorial design with
addition of Axial Points.
• Axial Points: +/- 1.414
• 1.414 = 2

31. Axial Points
• All points are equal distance from Center
(square root of 2)
• Leads to “Orthogonality” or stability of
design.
• Allow for estimate of individual curvature
effects.
• Central Composite is a very common type of
Response Surface Design
1.414

32. Part 5

33. Factorial Plus Axial Points (plus other
responses)
• Want to maximize yield and
also understand ideal Viscosity
and Molecular Weight.

34. Response Surface in MTB (Create RSM Table)
1. Select “Stat” Menu at top.
2. Select “DOE” platform.
3. Chose “Response Surface”
4. Chose “Create Response Surface Design”
1. Choose “Central Composite” for Type of Design.
2. This design has 2 continuous factors, and 0 categorical
factors.
3. Click “Designs…”
4. Click “Model” to select model specifics.

35. RSM Analysis in MTBesponse Surface in MTB
(Create RSM Table)
1. Under “Create Response Surface Designs”:
2. Number of Center Points is 5
3. Default Alpha (1.414) is correct.
4. No blocking in this case, only 1 replicate
5. Click “OK”
Will create basic data table with 2x2 factorial, 5
center points, and 4 axial points at 1.414.
Same as Factorial data set, but with Axial
points added in.

36. RSM Analysis in MTB
1. Select “Stat” Menu at top.
2. Select “DOE” platform.
3. Chose “Response Surface”
4. Chose “Analyze Response Surface Design”
1. Select the “Yield” column as the “Response”
2. Select “Terms” to set model parameters.

37. RSM Analysis in MTB
1. Selection of “Full quadratic” under “Include
the following terms” will set analysis for all
Main Effects, Quadratics, and Interactions.
2. Click “OK”
3. Click “OK” from Analysis window to run
analysis.

38. RSM Analysis in MTB
• Model shows strong statistical significance for
Quadratic and Interaction.
• Indicates a relative maximum is within this range.
• 3D Relationship can be determined with
Regression Equation.
• Maximum is point where partial derivatives of all
factors = 0.

39. Contour Plot
• Maximum Yield at approximately
point of darkest green in Contour
Plot.

40. Note: Need for Contour Examination
• Could be “peak” or
maximum
• Could be relative
maximum or “saddle
point

41. Maximum Yield
• Maximum Yield predicted at point
where coded Time = 0.3857, coded
Temp = 0.3
• Reminder, when (A, B) = (0, 0) then
(Time, Temp) = (85, 175) and +/- 1
in coded variables is +/- 5 in real
variables.

42. Maximum Yield
• Ideal Time = 85 + 0.3857*(5) = 86.9
• Ideal Temp = 175 + 0.3*(5) = 176.5
• IS this the best? Could be. Would
need to run confirmation runs to
determine.

43. Part 6

44. Multiple Responses Example
• Using this data, Chemical
Yield is most important.
Data also gathered on
Viscosity and Molecular
Weight.
• Goals:
• Maximize Yield
• Viscosity as close as
possible to 65
• Minimize Molecular Weight.

45. Contour Plots
• Examine Contour Plots

46. Response Surface Optimization in MTB
1. Select “Stat” Menu at top.
2. Select “DOE” platform.
3. Chose “Response Surface”
4. Chose “Response Optimizer”
1. Select goals for the responses (Options are to Maximize,
Minimize, or hit a Target).
2. “Options” can if needed set constraints on inputs.
3. “Setup” can if needed weight the importance of Outputs
4. Click “OK”

47. Response Optimization
• Will show how Responses will change relative
to Inputs.
• Desirability: Measure of combined strength of
optimal settings.
• Can go to “Interactive Mode”

48. Interactive Response Optimizer
• Can move red lines around to
see how these will interact.
• In ideal situation, this process is
a long collaborative process.
• Selection of ideal settings
should take into account many
perspectives, considerations.

49. Types of Designs to Fit a Response Surface
• Various designs exist to estimate the Response Surface.
• Some combination of
• Values at +/- 1
• Center Points
• Axial Points

50. Part 7

51. Central Composite Design
• Corner Points (A, B, C, … at +1 or -1)
• Center Points (A=B=C= … = 0)
• Axial Points (At +/- 𝛼)
• Is “Rotatable”, meaning all points at same
distance from center.
• Consistent variation of estimates.

52. Box-Behnken Design
• 3 level design
• No corner points
• Center Points
• Points on “side” of cube
• All points at consistent
radius
• Useful if corners or axial
points are at points
physically impossible.

53. Mixture Experiments
• An alloy is being created using some combination of lead and tin.
• It is possible to have between 40% to 60% lead (Pb) in the alloy. This
also means between 40 to 60% tin (Sn).
• Does this initial factorial experiment work?
Pb
Sn
60%
40%
40%
60%

54. Mixture Experiments
• Mixture levels constrained, that sum of percents will add up to 100%
• Increase of one level will lower the other levels.
0 ≤ 𝑥𝑖 ≤ 1, i = 1, 2, … , p
𝑥1 + 𝑥2 + ⋯ + 𝑥𝑝 = 1

55. Mixture Experiment Setup
• 2 factors: Straight
line
• 3 factors: Triangle
40 % Pb Content 60 %
Sn Content 40 %
60 %

56. Mixture Experiment Example
• 3 chemical components are combined
into a fiber to be spun into a yarn.
• Polyethylene (x1)
• Polystyrene (x2)
• Poly-propylene (x3)
• Response is yarn elongation strength.
• Simplex design chosen.
• Either pure blends or half

57. Experiment Results
• Statistical test for interactions
prioritized over main effects (since at
heart mixtures test interactions)
• Ideal setting:
X2=0
X3=80%
X1=20%