Uniform Design for Experiments with Mixtures

Abstract

This chapter introduces uniform design and modeling for experiments with mixtures and for experiments with restricted mixtures. Firstly, some designs for experiments with mixtures including the Scheffé simplex-lattice, simplex-centroid designs, and axial designs are introduced. Secondly, the uniform design of experiments with mixtures and the corresponding uniformity criteria are introduced. Finally, various modeling techniques for designs with mixtures are given.

Exercises

8.1

Give experimental points of the simplex-lattice designs $\{3,3\}, \{4,3\}$ and their plots by the use of MATLAB or other software.

8.2

The domain $T^3$ is an equilateral triangular with side-length $\sqrt{2}$, denoted by $V^2$, say. Therefore, any point $(z_1, z_2)$ on $V^2$ corresponds to a point $(x_1, x_2, x_3)$ on $T^3$. Choose a new coordinate system on $V^2$ and give the mapping rule of $(x_1, x_2, x_3)\Rightarrow (z_1, z_2)$.

8.3

Suppose we choose a uniform design $U_7(7^2)$ as follows:

Construct a uniform design on $T^3=\{(x_1,x_2, x_3): x_i> 0, i =1,2,3, x_1 + x_2 + x_3 = 1. \}$ with 7 runs by using the translation method based on the given $U_7(7^2)$.

8.4

Let $n=17$.

(1) :

Randomly choose n points on $[0,1]^2$ to form the design $D_1$ and calculate its mixture discrepancy.

(2) :

Use the translation method to obtain the design $D_2$ on $T^3$. Calculate the mean square distance, average distance, maximum distance of $D_2$.

 

No.	1	2
1	1	5
2	2	2
3	3	7
4	4	4
5	5	1
6	6	6
7	7	3

Repeat Steps (1)–(2) m times, compare their results, and give your conclusion.

8.5

Let $n=17$. Use the NTLBG algorithm to construct the uniform mixture designs on $T^3$.

8.6

For the designs with restricted mixtures, prove the restriction in (8.2.8).

8.7

Consider the three factors in Example 8.2.3. Use the conditional method to construct a 17-run uniform design with restricted mixtures.

8.8

Consider the design region

$$S_2=\{(x_1,x_2,x_3)|x_1+x_2+x_3=1, x_1^2+x_2^2\leqslant 0.36, x_i\geqslant 0\}.$$

Under the uniformity criterion CCD, use the switching algorithm in Algorithm 8.2.4 to construct a 15-point uniform design on $S_2$.

8.9

To explore the influence of component compatibility changes on antipyretic effect of Maxing Shigan decoction, the uniform design of experiments with mixtures was used. Ephedrae Herba ($x_1$/g), Armeniacae Semen Amarum ($x_2$/g), Glycyrrhizae Radix et Rhizoma Preparata Cum Melle ($x_3$/g), and Gypsum Fibrosum ($x_4$/g) were considered as 4 factors. The originally used treatment in hospitals is (6, 6, 6, 24), and the total weight is 42 g, and the response, the heat inhibition rate after 6 h, denoted by y(%), is equal to 52.19%. For investigating the reasonableness of the original treatment and finding better treatment, the researcher designed 12 different allocated proportions of Maxing Shigan decoction. The total weight of the four factors are kept to 42 g, and the corresponding design points and the response are as follows.

Analyze the data, compare the result of the original treatment, and give your conclusion.

No.	$x_1$(g)	$x_2$(g)	$x_3$(g)	$x_4$(g)	y
1	3.15	25.12	12.02	1.72	41.97
2	17.1	19.82	2.33	2.75	36.13
3	21	2.32	3.89	14.79	28.47
4	1.83	15.57	7.17	17.42	52.92
5	0.59	6.56	18.88	15.97	53.28
6	11.71	16.46	1.73	12.1	29.93
7	14.15	5.83	21.1	0.92	16.79
8	6.09	2.32	26.59	7	29.1
9	7.76	15.75	11.56	6.93	49.64
10	4.56	9.88	1.15	26.41	56.75
11	9.62	0.68	11.89	19.81	52.19
12	27.44	4.7	6.98	2.88	10.53

8.10

In an experiment for Chinese medicinal material, five components are considered and the restricted ranges of the components $x_1\sim x_5$ are 10%$\sim$60%, 10% $\sim$60%, 30%$\sim$60%, 10% $\sim$12%, 10% $\sim$ 12%, respectively. The average yield (g) and survival rate (%) are two responses and denoted by $y_1$ and $y_2$.

Analyze the data and find the optimal components.

No.	$x_1$(%)	$x_2$(%)	$x_3$(%)	$x_4$(%)	$x_5$(%)	$y_1$	$y_2$
1	15.66	36.69	45.31	0.98	1.36	284.5	44.44
2	33.89	16.77	41.14	1.66	6.53	356.8	44.44
3	19.77	19.03	57.39	1.69	2.11	337.9	88.89
4	36.21	13.36	32.77	8.28	9.37	463.8	100
5	47.08	16.54	33.83	1.28	1.27	326.3	66.66
6	15.57	39.57	35.26	0.65	8.95	454.3	100
7	20.95	33.3	35.23	4.42	6.1	359.1	88.89
8	38.23	14.16	45.3	1.45	0.85	381.4	55.56
9	40.21	16.78	33.03	0.68	9.3	446	55.56
10	17.32	18.97	52.68	0.97	10.05	433.3	77.78
11	18.57	16.08	54.43	9.88	1.04	342.7	66.67
12	31.03	28.74	33.45	5.66	1.12	374	55.56
13	15.96	40.91	32.56	9.72	0.85	397	44.44
14	13.05	34.97	33.66	9.27	9.05	416	88.89
15	14	14.02	52.72	9.33	9.92	475.9	100
16	14.72	33.72	42.64	7.91	1.01	290	22.22
17	32.14	32.36	33.02	0.6	1.88	317.4	88.89
18	40.5	13.49	36.32	9.03	0.66	349.8	44.44
19	26.26	20.98	38.29	8.7	5.77	474.25	44.44
20	15.57	48.33	33.9	1.14	1.06	0	0