(Response Surface Methodology)

1
Chapter 11

• Part I ??????
• (Response Surface Methodology)

2
Why/When to Use RSM?

• ???????(Response Variable)????????.
• ???????????.
• ??????????????
• ???
• ???
• ???
• ?? ?????????(??), ????? ?????.

3
RSM?????

• ??????? Y f(x1, x2) e
• ????(Response Surface) ? f(x1, x2)
• ????????, ? f(x1, x2) ????????. ?
• Y b0b1x1b2x2bkxke (first order)
• Y b0?bixi?biix2i? ? bijxixje (second
  order)

4
???? - Example
5
The Method of Steepest Ascent

• ?? ??????????????????.
• ?? ?????????????, ????, ?? First-order Model
  ????.
• Steepest Ascent ???????????(?????????????),
  ?????????.
• ???????, ??? Steepest Descent.

6
Steepest Ascent - ??
7
Steepest Ascent - Example

• 525.DX5
• ?? ?1 ???? (35 min.)
• ?2 ???? (155 oF)
• ???? Y ?????? (40)
• Coded Variable (X1X2) (-1 1 -1 1)
• Natural Variable (? 1 ? 2) (30 40 150
  160)

8
Example 525 ?????

• ?????
• Error ???
• First-order Model
• ???? ( Fit? )

9
Example ? ANOVA Table
10
Example?????

• ?????????(Regression Model)?
• y 40.44 0.775x1 0.325x2
• x1?x2???(0.775 and 0.325)??????standard error
  sqrt(MSE/d.f.e) 0.10??? ???????.
• ?????????
• ?????????????? (?Coded Variable ???), ??? ?x1
  1, ??x2 (0.325/0.775) ?x1 0.42

11
Example ???????(?)
12
Example ???????(?)
13
Example ???????(?)-ANOVA

• ?????????(Regression Model)?
• y 78.97 1.00x1 0.50x2
• ?????????????.

14
Steepest Ascent ??

• 2k nc center point ? CCD ? ??
• First-order Model ??, ?Curvature??? ?????????.
• ?????????? ???Step Size ?xi.
• ?????Step Size gt
• ?xi / bi ?xk / bk
• ??xi???Natural Variable ??????.

15
Second-order Model ???

• ?????????, First-order Model????? ????
  Second-order Model ?????Model????????????(??)??.

16
Central Composite Design (CCD) - Example

• 534.DX5

17
CCD ???
18
CCD Example ? ANOVA
19
CCD Example ?????
20
CCD Example ?????_Contour Plot
21
Chapter 11

• Part II ??????
• - ?????
• - Optimization
• - EVOP

22
?????????????

• ??????, ??????????
• ?? Model ?????? (Lack of Fit)
• ????? (Blocking)
• ???? Model ???????????
• ?????? (Pure Error) ???
• ???????
• ????????
• ?? Model ????????????

23
?? Model ? RSM ??

• ???? ?? (Othogonal)
• 2k nc center point
• 2k-p nc center point, ???????III??, Why?
• Simplex Design
• k ???, ?? k1 ?(??)??

24
Simplex Design
25
????? - Example

• 23
• ???? Pure Error - 4 d.f. ? Lack-of-fit
• ?? Model????????
• 23-1 4 center point
• 3 d.f. ? Pure Error - 1 d.f. ?
  Curvature
• ?? ????????
• 23-1, n 2
• 4 d.f. ? Pure Error - ????
  Lack-of-fit
• ?? ????????????
• ???23 4 center point

26
?? Model ? RSM ??(1/4)

• ???? ?? (Orthogonal) ? ???? (Rotatable)
• Central Composite Designs (CCDs)
• 2k ?2k-1 (???V) 2k ??? (Axial Points) nc
  center point
• Factoral Points 2k?2k-1 (???V) ?????????????
• Axial Points ????????
• Center Points????????? Pure Error

27
Information Surfaces and Contours_22 Design
28
Information Surfaces and Contours_32 Design
29
Information Surfaces and Contours_Second-order
Rotatable Design
30
CCD ??
31
??? CCDs
32
(No Transcript)
33
?? Model ? RSM ??(2/4)

• Face-centered Central Composite Design (FCCD)

• ?? a 1??, ??? CCDs?
• ?????????????, ???????
• ???? (Rotatability) ??, ???????

34
?? Model ? RSM ??(3/4)

• Box-Behnken Design
• ???????? (-1, 0, 1)
• ??????22, ???????????, ??? nc center points
• Rotatability ? CCDs ????, ????
• ? k 3 ?, ????? CCD ??? 12nc Vs. 14nc.
• ? k 4 ?, ????? CCD ? gt 24nc.
• ? k 5 ???, ????? CCD ???.

35
Box-Behnken Design (k 3)
36
Box-Behnken Design (k 4, 5)
37
?? Model ? RSM ??(4/4)

• Hybrid Designs
• ?? k-1 ????????? CCDs, ???? (kth) ???????????????
• ????? (Small Sample Size)
• ????? k 3,4,6,7

38
Hybrid Design ??

• Hybrid 310 Hybrid 311A

39
Hybrid Design ??

• Hybrid 416A, 416B, 416C

40
Design Optimality Criteria

• D-Optimality and D-Efficiency
• Rotatability
• Model ?????????
• A-Optimality
• Model ????????
• G- and Q-Optimality
• ? Model ???????????

41
????? RSM Designs
42
Evolutionary Operation (EVOP)

• ?????????????????????????????, ??????, ?????????
  (drift). ?????????????????.
• EVOP ????????, ???????, ??????????,
  ????????????????.
• 2k center point, ? cycle ?????.

43
EVOP ???
44
EVOP Example (1/5)
45
EVOP Example (2/5)
46
EVOP Example (3/5)
47
EVOP Example (4/5)
48
EVOP Example (5/5)
49
Chapter 11

• Part III ????
• (Mixture Designs/Experiments)

50
???????

• ?????????????, ????????????????? (Independent)
  ??, ??????, ?????????????, ????????????????????
  (Dependent). ??, ??????????? (Mixture Designs)
  ???????????.
• Example
• ??/????????????.

51
??????????

• ?? x1,x2,...,xp ???????????????, ?
• 0 ? xi ? 1, i 1, 2, ..., p
• ?
• x1 x2 ... xp 1 (i.e. 100)

52
???????
53
???????? (Trilinear Coordinate System)
54
p, m Simplex Lattice Design

• p ???, ????? m 1 ???.

55
Simplex Centroid Design

• p ???? 2p-1 ???

56
Mixture Models

• Linear
• Y ?bixi
• Quadratic
• Y ?bixi ? ?bijxi xj
• Cubic
• Y ?bixi ? ?bijxi xj ? ? sijxixj(xi -
  xj)
• ? ? ?bijkxi xj xk
• Special Cubic
• Y ?bixi ? ?bijxi xj ? ? ?bijkxi xj xk

57
Example 556.DX5

• ?? ??????, ????, ????.
• x1 ???
• x2 ????
• x3 ???
• ????
• y ???

58
??????

• Screening Design
• ??????, ??????????.
• D-Optimal Design
• ????
• Distance based
• ?????
• Modified distance
• ?????

59
???????
??????????????, ?????, ???????.
??????
Screening Experiments
- ?????? - 2k-pnc ???? III (?)??.
2k-pnc
2knc
Fold Over
- ???????
-???????
-???????
One Factor
- ??????
60
?????????
????
- ??????
- ?????, ?????????????. - 2knc, ? 2k-pnc ????
III ??.
Steepest Ascent
?????
CCD
Box-Behnken
Hybrid
- ??????? 3
- ???? 3, 4, 6. - ??????
- ??????? 5
61
???????
Screening Design
????
Simplex Lattice Design
Simplex Centroid Design
??????