
After experimentation has been completed, and a process or design has been
optimized, it is not uncommon for the experimentation team to consider the
problem to be solved. In many cases, the optimum solution does not stay optimum;
equipment changes as it ages, the composition of components change over time,
and many other factors make small changes that affect the response variable.
Evolutionary Operation (EVOP) is a method of addressing the constant small
changes in a process in attempt to maintain an optimum level. Experimentation is
carried out on a near continuous basis by introducing small changes to the
system.
One of the most popular methods of EVOP is sequential simplex optimization.
The simplex is a geometric figure with the number of vertexes equal to the
number of factors plus one. A simplex with one factor is a line, a simplex with
two factors is a triangle, etc. The simplex is a simplistic model of the
surface. a new simplex is formed by eliminating the vertex with the worst
response and replacing it by projecting it through the average coordinates of
the remaining vertexes. A new experiment is performed with the factor levels
determined by the coordinates of the new vertex, and the process is repeated
until an optimum response is found. Using the sequential simplex method
optimization has two advantages over using factorial designed experiments for
EVOP.
 The number of trials in the initial simplex is k+1 (k is the number
of experimental factors), while a factorial approach has at least 2^{k}
and possibly 3^{k} or 4^{k}.
 Only one new trial is required to move to a new area in the space defined
by the factors; while a factorial design requires at least 2^{k1}
trials. An example of a sequential simplex optimization is shown in the
figure below.
To compute the coordinates of the new vertex (R), refer to the following
table, where B is the vertex with the best response, N is the vertex with the
next best response, and W is the vertex with the worst response. This table
shows an example for a 3 factor optimization (k = 3). The logic is the
same for any number of factors.

A

B

C

Result

Next Best

20

20

20

425

Best

20

30

20

503

Other

30

20

20

378

Sum

70

70

60


Sum/k

23.3

23.3

20


Worst

20

20

15

215

Sum/kW

3.3

3.3

5


R

26.7

26.7

25


(Sum/kW)/2

1.7

1.7

2.5


Cw

21.7

21.7

17.5


Cr

25

25

22.5


E

30

30

30


The optimum point is never reached in the figure above because the size of
the individual simplex is too large. If the simplex size is too small, an
excessive number of steps is required to reach the optimum point. This problem
has been solved by using a variable size simplex. Instead of a simple reflection
(R), there are other options:
 double the length of the reflection (E),
 reduce the length of the expansion by 50% (Cr),
 produce the reflection in the opposite direction (W), and
 produce the reflection in the opposite direction but at 50% of the normal
length (Cw).
The additional rules for determining the location of the new vertex are:
 If the response for the new vertex (R) is better than the response for the
next best vertex (N), but worse than the response of the best (B) vertex,
then use R as the new vertex.
 If the response for the new vertex (R) is better than the response of the
best vertex (B), then compute E = P + 2(PW).
 If the response of E is better than the response of B, use E as the new
vertex; if not, use R as the new vertex.
 If the response for the new vertex (R) is worse than the response of the
next best vertex (N) but better than the response of the worst vertex (W),
then compute Cr = P + 0.5(PW) as the new vertex.
 If the response for the new vertex (R) is worse than the response of the
worst vertex (W), then compute Cw = P  0.5(PW) as the new vertex.
Example
Consider the parameter Y determined from the function Starting with the
vertices
Y = 40A + 35B  15A^{2}  15B^{2} + 25AB
Solution
Starting with the vertices
Vertex 
A 
B 
Response 
1 
100 
100 
42,500 
2 
100 
120 
57,800 
3 
120 
120 
63,000 
The response Y is maximized using a variablestep sequential simplex as shown
below.
Start






Coordinates 
A 
B 
Response 
Rank 

Best
(B) 
100 
100 
42500 
B 

Next
Best (N) 
100 
120 
57800 
N 

Sum 
200 
220 



P bar 
100 
110 



Worst (W) 
120 
120 
63000 
W 

P bar  W 
20 
10 



Next Vertex (R) 
80 
100 
39300 
R 
Compute
E 
E
= 
60 
90 
34950 

Use
E 






Step 2






Coordinates 
A 
B 
Response 
Rank 

Best
(B) 
60 
90 
34950 
B 

Next
Best (N) 
100 
100 
42500 
N 

Sum 
160 
190 



P bar 
80 
95 



Worst (W) 
100 
120 
57800 
W 

P bar  W 
20 
25 



Next Vertex (R) 
60 
70 
17650 
R 
Compute
E 
E
= 
40 
45 
6200 

Use
E 






Step 3






Coordinates 
A 
B 
Response 
Rank 

Best
(B) 
40 
45 
6200 
B 

Next
Best (N) 
60 
90 
34950 
N 

Sum 
100 
135 



P bar 
50 
67.5 



Worst (W) 
100 
100 
42500 
W 

P bar  W 
50 
32.5 



Next
Vertex (R) 
0 
35 
17150 
R 
Use
R 






Step 4






Coordinates 
A 
B 
Response 
Rank 

Best
(B) 
40 
45 
6200 
B 

Next
Best (N) 
0 
35 
17150 
N 

Sum 
40 
80 



P bar 
20 
40 



Worst (W) 
60 
90 
34950 
W 

P bar  W 
40 
50 



Next Vertex (R) 
20 
10 
3650 
R 
Compute
E 
E
= 
60 
60 
22500 

Use
R 






Step 5






Coordinates 
A 
B 
Response 
Rank 

Best
(B) 
20 
10 
3650 
B 

Next
Best (N) 
40 
45 
6200 
N 

Sum 
20 
35 



P bar 
10 
17.5 



Worst (W) 
0 
35 
17150 
W 

P bar  W 
10 
17.5 



Next Vertex (R) 
20 
0 
5200 
R 
Use
R 






Step 6






Coordinates 
A 
B 
Response 
Rank 

Best
(B) 
20 
10 
3650 
B 

Next
Best (N) 
20 
0 
5200 
N 

Sum 
0 
10 



P bar 
0 
5 



Worst (W) 
40 
45 
6200 
W 

P bar  W 
40 
50 



Next Vertex (R) 
40 
55 
17900 
R 
Compute
Cw 
Cw
= 
20 
20 
500 

Use
Cw 






Step 7






Coordinates 
A 
B 
Response 
Rank 

Best
(B) 
20 
20 
500 
B 

Next
Best (N) 
20 
10 
3650 
N 

Sum 
0 
10 



P bar 
0 
5 



Worst (W) 
20 
0 
5200 
W 

P bar  W 
20 
5 



Next Vertex (R) 
20 
10 
12950 
R 
Compute
Cw 
Cw
= 
10 
2.5 
481 

Use
Cw 






Step 8






Coordinates 
A 
B 
Response 
Rank 

Best
(B) 
10 
2.5 
481 
B 

Next
Best (N) 
20 
20 
500 
N 

Sum 
30 
22.5 



P bar 
15 
11.25 



Worst (W) 
20 
10 
3650 
W 

P bar  W 
35 
21.25 



Next Vertex (R) 
50 
32.5 
9581 
R 
Compute
Cw 
Cw
= 
2.5 
0.625 
217 

Use
Cw 






Step 9






Coordinates 
A 
B 
Response 
Rank 

Best
(B) 
2.5 
0.625 
217 
B 

Next
Best (N) 
10 
2.5 
481 
N 

Sum 
7.5 
3.125 



P bar 
3.75 
1.5625 



Worst (W) 
20 
20 
500 
W 

P bar  W 
16.25 
18.4375 



Next Vertex (R) 
12.5 
16.875 
2432 
R 
Compute
Cw 
Cw
= 
11.875 
10.78125 
194 

Use
Cw 






Step 10






Coordinates 
A 
B 
Response 
Rank 

Best
(B) 
11.875 
10.78125 
194 
B 

Next
Best (N) 
2.5 
0.625 
217 
N 

Sum 
9.375 
11.40625 



P bar 
4.6875 
5.703125 



Worst (W) 
10 
2.5 
481 
W 

P bar  W 
5.3125 
3.203125 



Next Vertex (R) 
0.625 
8.90625 
1048 
R 
Compute
Cw 
Cw
= 
7.34375 
4.101563 
129 

Use
Cw 






Step 11






Coordinates 
A 
B 
Response 
Rank 

Best
(B) 
11.875 
10.78125 
194 
B 

Next
Best (N) 
7.34375 
4.101563 
129 
N 

Sum 
19.21875 
14.88281 



P bar 
9.609375 
7.441407 



Worst (W) 
2.5 
0.625 
217 
W 

P bar  W 
12.10938 
6.816407 



Next Vertex (R) 
21.71875 
14.25781 
1016 
R 
Compute
Cw 
Cw
= 
3.554688 
4.033203 
208 

Use
Cw 






Step 12






Coordinates 
A 
B 
Response 
Rank 

Best
(B) 
3.554688 
4.033203 
208 
B 

Next
Best (N) 
11.875 
10.78125 
194 
N 

Sum 
15.42969 
14.81445 



P bar 
7.714844 
7.407227 



Worst (W) 
7.34375 
4.101563 
129 
W 

P bar  W 
0.371094 
3.305664 



Next Vertex (R) 
8.085938 
10.71289 
162 
R 
Compute
Cr 
Cr
= 
7.900391 
9.060058 
255 

Use
Cr 






Step 13






Coordinates 
A 
B 
Response 
Rank 

Best
(B) 
7.900391 
9.060058 
255 
B 

Next
Best (N) 
3.554688 
4.033203 
208 
N 

Sum 
11.45508 
13.09326 



P bar 
5.727539 
6.546631 



Worst (W) 
11.875 
10.78125 
194 
W 

P bar  W 
6.14746 
4.23462 



Next Vertex (R) 
0.41992 
2.312012 
43 
R 
Compute
Cw 
Cw
= 
8.80127 
8.66394 
274 

Use
Cw 






Step 14






Coordinates 
A 
B 
Response 
Rank 

Best
(B) 
8.80127 
8.66394 
274 
B 

Next
Best (N) 
7.900391 
9.060058 
255 
N 

Sum 
16.70166 
17.724 



P bar 
8.35083 
8.861999 



Worst (W) 
3.554688 
4.033203 
208 
W 

P bar  W 
4.796143 
4.828796 



Next Vertex (R) 
13.14697 
13.6908 
101 
R 
Compute
Cw 
Cw
= 
5.952759 
6.447601 
268 

Use
Cw 






Step 15






Coordinates 
A 
B 
Response 
Rank 

Best
(B) 
8.80127 
8.66394 
274 
B 

Next
Best (N) 
5.952759 
6.447601 
268 
N 

Sum 
14.75403 
15.11154 



P bar 
7.377014 
7.555771 



Worst (W) 
7.900391 
9.060058 
255 
W 

P bar  W 
0.52338 
1.50429 



Next Vertex (R) 
6.853638 
6.051483 
269 
R 
Use
R 
R
= 
6.853638 
6.051483 
269 

Use
R 






Step 16






Coordinates 
A 
B 
Response 
Rank 

Best
(B) 
8.80127 
8.66394 
274 
B 

Next
Best (N) 
6.853638 
6.051483 
269 
N 

Sum 
15.65491 
14.71542 



P bar 
7.827454 
7.357712 



Worst (W) 
5.952759 
6.447601 
268 
W 

P bar  W 
1.874695 
0.910111 



Next Vertex (R) 
9.702148 
8.267822 
246 
R 
Compute
Cw 
Cw
= 
6.890106 
6.902657 
279 

Use
Cw 
After 16 step the best response is 279. The optimum value is 281 when A =
7.54 and 7.46.
