Otimização Simplex em Química Analítica

Prévia do material em texto

230 Solutions Manual for Analytical Chemistry 2.1
labeled “simplex” shows the three vertices that make up each simplex. 
For each simplex, the vertex that we reject is shown in bold font; note 
that on two occasions, the rejected vertex, shown in bold-italic font, 
has the second-worst response (either because of a boundary condi-
tion or because the new vertex has the worst response)
vertex A B response simplex
1 0 0 3.264 —
2 1.0 0 4.651 —
3 0.5 0.87 4.442 1, 2, 3
4 1.5 0.87 5.627 2, 3, 4
5 2.0 0 5.736 2, 4, 5
6 2.5 0.87 6.512 4, 5, 6
7 3.0 0 6.521 5, 6, 7
8 3.5 0.87 7.096 6, 7, 8
9 4.0 0 7.004 7, 8, 9
10 4.5 0.87 7.378 8, 9, 10
11 4.0 1.74 7.504 8, 10, 11
12 5.0 1.74 7.586 10, 11, 12
13 4.5 2.61 7.745 11, 12, 13
14 5.5 2.61 7.626 12, 13, 14
15 5.0 3.48 7.820 13, 14, 15
16 4.0 3.48 7.839 13, 15, 16
17 4.5 4.35 7.947 15, 16, 17
18 3.5 4.35 7.866 16, 17, 18
19 4.0 5.22 8.008 17, 18, 19
20 5.0 5.22 7.888 17, 19, 20
21 4.5 6.09 7.983 19, 20, 21
22 3.5 6.09 8.002 19, 21, 22
23 3.0 5.22 7.826 19, 22, 23
24 3.5 4.35 7.866 19, 23, 24
25 4.5 4.35 7.947 19, 24, 25
 Figure SM14.4 shows the progress of the simplex optimization in 
three dimensions and in two dimensions.
3. To help us in the derivation, we will use the diagram shown in Figure 
SM14.5 where a and b are the coordinates of a vertex, and w, b, s, and 
n identify the vertex with, respectively, the worst response, the best 
response, the second-best response, and the new vertex. he red cir-
cle marks the midpoint between the best vertex and the second-best 
vertex; its coordinates are
values of a va
lu
es
 o
f b
re
sp
o
n
se
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
values of a
v
a
lu
e
s 
o
f 
b
0 2 4 6 8 10
0
2
4
6
8
1
0
Figure SM14.4 Two views showing the 
progress of a simplex optimization of the 
equation in Problem 1c in (a) three dimen-
sions and in (b) two dimensions. he leg-
end shows the colors used for the individ-
ual contour lines. Figure SM14.3b shows 
the full response surface for this problem.
(ab, bb)
(an, bn)
(as, bs)
(aw, bw)
a
b
Figure SM14.5 Diagram showing vertices 
of original simplex and the relection of the 
worst vertex across the midpoint (red cir-
cle) of the best and the next-best vertices to 
give the new vertex (green circle). See text 
for additional details.