Response Surface Methodology

1
Response Surface Methodology

• ? ? ?
• ???????????
• ?????????
• ???????? ????

2

• Introduction
• Many experimental programs are designed with a
  two-fold purpose in mind to quantify the
  relationship between the values of some
  measurable response variable(s) and those of a
  set of experimental factors presumed to affect
  the response(s) and, to find the values of the
  factors that produce the best value or values of
  the response(s).

3

• Example 1
• Upon ascertaining that the purity of an end
  product of a chemical process is affected by the
  concentrations of several reagents in the
  solution as well as the temperature of the
  reaction, the chemist would then try to determine
  the specific concentrations of the reagents and
  the temperature level that results in the highest
  degree of purity of the product.

4

• Example 2
• Suppose a series of clinical trials is conducted
  on patients with high blood pressure where the
  patients are given selected combinations of two
  drugs each known to reduce their blood pressure
  reading. Here again the purpose of administering
  several combinations of drugs to the various
  individuals is to find the specific combination
  of drugs that reduces the patients blood
  pressure reading by the greatest amount during
  some specified interval of time.

5

• Note
• Response surface methodology (RSM) is a set of
  techniques designed to find the best value of
  the response. If discovering the best value or
  values of the response is beyond the available
  resources of the experiment, then response
  surface methods are used to at least gain a
  better understanding of the overall response
  system.

6

• Illustrative Example
• Response (Y ) Yield
• Input Variable (X1) Time
• Input Variable (X2) Temperature

Note The initial set of experiments consists of
looking at two levels of time (70 min and 80
min) and two levels of temperature (127.5o and
132.5oC).
7
Step 1 The initial set of experiments consists
of looking at two levels of time (70 min and 80
min) and two levels of temperature (127.5o and
132.5oC).
8
(No Transcript)
9
???
10
Conclusion First-order Model is adequate.
Results of Least Square Fitting
Action Taken Steepest Ascent ---
direction for improvement --- 2.35 4.5 (or
1 1.91)
11
Step 2 Run more experiments Following the
direction of 1 1.91 (2.35 4.5)
12
Temperature
160
150
140
x
x
x
130
x
x
120
Time
60
70
80
90
100
110
13
Temperature
1 1.91
160
58.2 R9
150
86.8 R10
140
Use Point 10 (R10) as the new center point and
start all over again!
73.3 R8
x
x
x
130
x
x
120
Time
60
70
80
90
100
110
14
Temperature
160
150
86.8 R10
140
Use Point 10 (R10) as the new center point and
start all over again!
x
x
x
130
x
x
120
Time
60
70
80
90
100
110
15
Step 3 Start all over again!! ---Use Point 10
(R10) as the new center point
16
(No Transcript)
17
t-test with 1 d.f. ? Need more points
18
Step 3 Start all over again!! ---Use Point 10
(R10) as the new center point
19
Step 4 Add few more points for fitting a
second-order model.
145
145
20
Temperature
160
79.5
77.4
91.2
150
89.7 86.8 87.0 86.0
83.3
81.2
140
78.8
84.5
81.2
x
x
x
130
x
x
120
Time
60
70
80
90
100
110
21
(No Transcript)
22
(No Transcript)
23
Temperature (oC)
Time (min)
24

• The global optimum turns out to be
• x180 minutes
• x2150 oC
• E(y)91.2

25
(No Transcript)