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

## Education Article

• Published: Jan 1, 2000
• Channels: Chemometrics & Informatics

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 0.3 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 xnew,f = xold,f (U,f - Lf) + Lf to give

 0.3 0.3 0.4 0 0.6 0.4 0 0.3 0.7 0.2 0.4 0.4 0.2 0.3 0.5 0.1 0.5 0.4 0 0.5 0.5 0.1 0.3 0.6 0 0.4 0.6 0.1 0.4 0.5

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