Solution 2.4 - Chemometrics: Data Analysis for the Laboratory and Chemical Plant

1. The equation is given on page 89 of the printed text. The numbers of experiments are 5, 15 and 35 for the three designs.

2. The full {5,3} simplex lattice design is given below.

The first five experiments consist of all the combinations of 3/3 (=1). The next twenty experiments consist of all the combinations of 2/3 and 1/3. The final ten experiments consist of all the combinations of 1/3, 1/3 and 1/3.

3. There are 2⁵-1 or 31 experiments altogether.

The design is as follows.

Note that there are

• 5 single component blends,
• 10 two component blends,
• 10 three component blends
• 5 four component blends,
• 1 five component blend.

Note also that the one component and three component blends correspond to identical experiments in the {5,3} simplex lattice design.

4. The design is given in Table 2.37 in the printed text and is as follows.

1	0	0
0	1	0
0	0	1
2/3	1/3	0
2/3	0	1/3
1/3	2/3	0
0	2/3	1/3
1/3	0	2/3
0	1/3	2/3
1/3	1/3	1/3

The first step is to establish upper bounds (see Table 2.39 in the printed text for an example) so that the new upper and lower bounds become

L	0.0	0.3	0.4
U	0.3	0.6	0.7

where the upper bounds of x₁ equals 1 minus the sum of the lower bounds of x₂ and x₃ (=1-0.3-0.4) and so on. Then the new mixture design can be obtained by transforming the original values by the equation x_new,f = x_old,f (U_,f - L_f) + L_f to give

0.3	0.3	0.4
0.0	0.6	0.4
0.0	0.3	0.7
0.2	0.4	0.4
0.2	0.3	0.5
0.1	0.5	0.4
0.0	0.5	0.5
0.1	0.3	0.6
0.0	0.4	0.6
0.1	0.4	0.5