One of important purpose in technical and scientific experiments is optimization of response, it means that we have to find out the values of factors `X_i` in order that the corresponding response `Y` attain the optimal value. To obtain this purpose, we can use full factorial experiments. But when the number of factors increases, the numbers of treatments and runs increase considerably, require important resources and sometimes impossible to realize.

But we recognize that to optimize the response, we can construct an empirical equation relating factors `X_i` and response `Y`:

`y=f(x_1,x_2,\ ...\ )`(14)

From (14), we can use mathematical methods to find out the value of factor `X_i` to optimize response `Y`.

To realize it, theory proved that we need only a moderate number of runs. And this is the basic of response surface methodology.

 

The relation between `Y` and `X_i` can be presented graphically by a surface, known as response surface (Fig.1).

Fig. 1 Response surface

So the methods in this strategy is denoted as response surface methodology (RSM).

To construct empirical equation (14), we can choose many forms of suitable functions. But the quadratic equation is the most widely used because it requires moderate numbers of treatments and runs, satisfies requirements of design and analysis of experiment, realizes easier in comparison with others way, conforms well to actual data, illustrates easily, ...

To determine the empirical equation, least squares method is used.

Among methods of response surface, central composite design and Box-Behnken method are the most frequently used.

 

An experiment that study the effects of `k` factors by central composite design (CCD) method consists of three groups of treatment:

• Full factorial treatments, in which each treatment is studied at two levels. There are `2^k` treatments for `k` factors.
• Axial treatments (or star treatment): for each treatment in this group, there is (`k–1`) factors having coded value 0, the last factor has coded value `-alpha` or `alpha`. There is `2k` treatments in this group.
• Center treatment : coded values of all factors are 0.

An experiment of CCD methods with 2 factors `X_1` and `X_2` is illustrated in Fig. 2.

Fig. 2 Illustration for CCD with 2 factors

Note

For the treatments of group 1 and 2, it is not necessary to replicate. But center treatment must be realized with `n` runs. Therefore an experiment of CCD method with `k` factors consists of `2^k+2k+1` treatments with at least `2^k+2k+n` runs.

The matrix of coded factor for a CCD experiment with 2 factors and 5 runs at center treatment is shown in Table 1.

Table 1 Matrix of coded factor of CCD experiment with 2 factors

Run	`X_1`	`X_2`
1	+	+
2	+	−
3	−	+
4	−	−
5	`alpha`	0
6	`-alpha`	0
7	0	`alpha`
8	0	`-alpha`
9	0	0
10	0	0
11	0	0
12	0	0
13	0	0

From Table 1, we recognize that:

• there are 4 runs for 4 full factorial treatments : they are runs numbers 1, 2, 3, 4.
• there are 4 runs for 4 axial treatments : they are runs numbers 5, 6, 7, 8.
• there are 5 runs for center treatment : they are runs numbers 9, 10, 11, 12, 13.

The coded value of `alpha` depends on characteristic of experiment (orthogonality, rotatability, ...) and actual context.

 

Box-Behnken method is used for experiments with more than 2 factors. Except center treatment, in each treatment of this method:

• `b` factors are at higher level (coded as + 1 or +) or lower level (coded as - 1 or −),
• other factor at center level (coded as 0).

In center treatment, the coded value of all factors is 0.

For example, matrix of coded factors for a Box-Behnken experiment with 3 factors is shown on Table 2.

Table 2 Matrix of coded factor for a Box-Behnken experiment with 3 factors

Run	`X_1`	`X_2`	`X_3`
1	− 1	0	1
2	0	− 1	1
3	1	0	1
4	0	1	1
5	− 1	1	0
6	− 1	− 1	0
7	1	− 1	0
8	1	1	0
9	− 1	0	− 1
10	0	− 1	− 1
11	1	0	− 1
12	0	1	− 1
13	0	0	0
14	0	0	0
15	0	0	0

We can recognize that besides runs 13, 14, 15 (center treatment), all the others run have one factor at level 0 and two other at level + 1 or - 1.