Box-Behnken design

A Box-Behnkens desing is test array specially designed for response surface methodology.

In the Box-Behnken design the Levels of the factors are at the midpoints of the edges () and in the center the Center point (). This array is often used in a RSM to construct a response surface. For each factor are 3 levels required.

Box-Behnken compared with Central Composite

As you can see there are less data point needed compared with a Central Composite design. Compared to the Central Composite design it is missing the corner points, it will never occur that all the factors are high or low at the same time (No extreme combinations).
With a Box-Behnken design every factor is having 3 levels compared to Central Composite 5 with 2 of them outside the setting.

Box-Behnken design	Central Composite design
Compare
-No Corner points	+Corner points
+3 Levels	-5 Levels
+Less tests	-more tests
+No Circumscribed points	-The Circumscribed point especially for the bigger arrays are far from the normal setting

Box-Behnken compared Full Factorial Array 3 Levels

Box-Behnken design	Full Factorial Array 3 Levels
Compare
-No Corner points	+Corner points
+Less tests	-more tests
+Rotatable	-Not Rotatable
-Always 3 Levels	+Flexible in the amount of levels
+Multiple Center points	+/-One Center points But they can easy be add

The arrays

Name	Factors	Rows	center points
BB3	3	15	3
BB4	4	27	3
BB5	5	45	5
BB6	6	54	6

BB3

Tests↓	Factors
	A	B	C
1	-1	-1	0
2	1	-1	0
3	-1	1	0
4	1	1	0
5	0	0	0
6	0	-1	-1
7	0	-1	1
8	0	1	-1
9	0	1	1
10	0	0	0
11	-1	0	-1
12	-1	0	1
13	1	0	-1
14	1	0	1
15	0	0	0

Graphical image of the BB3 array

BB4

Tests	Factors
	A	B	C	D
1	-1	-1	0	0
2	-1	1	0	0
3	1	-1	0	0
4	1	1	0	0
5	0	0	-1	-1
6	0	0	-1	1
7	0	0	1	-1
8	0	0	1	1
9	0	0	0	0
10	-1	0	0	-1
11	-1	0	0	1
12	1	0	0	-1
13	1	0	0	1
14	0	-1	-1	0
15	0	-1	1	0
16	0	1	-1	0
17	0	1	1	0
18	0	0	0	0
19	-1	0	-1	0
20	-1	0	1	0
21	1	0	-1	0
22	1	0	1	0
23	0	-1	0	-1
24	0	-1	0	1
25	0	1	0	-1
26	0	1	0	1
27	0	0	0	0

Links

• http://www.itl.nist.gov/div898/handbook/pri/section3/pri3362.htm
• https://en.wikipedia.org/wiki/Box%E2%80%93Behnken_design