# Sequential Response Surface Methods

**Creating 3D Surface Plots in Excel:**

- The most important outcomes of an RSM application are often 3D surface plots. This follows because they are readily interpretable by a wide variety of people and help in building intuition about the system studied. Yet these plots are
- inappropriate for cases in which only first order terms have large coefficients.
- Creating a contour plot in Excel requires manual creation of formulas to generate an array of predictions needed by the Excel 3D charting routine to create the plot.
- The dollar signs are selected such that the formula in cell E8 can be copied to all cells in the range E8:J20, producing correct predictions. Having generated all the predictions and putting the desired axes values in cells, E7:J7 and D8:D20, then the entire region D7:J20 is selected, and the “Chart” utility is called up through the “Insert” menu.

**Sequential Response Surface Methods:**

**Definition**: “**Block**” here refers to a batch of experimental runs that are performed at one time. Time here is a blocking factor that we cannot randomize over. Rows of experimental plans associated with blocks are not intended to structure experimentation for usual factors or system inputs. If they are used for usual factors, then prediction performance may degrade substantially.

**Definition**: A “**center point**” is an experimental run with all of the settings set at their mid-value. For example, if a factor ranges from 15” to 20” in the region of interest to the experiment, the “center point” would have a value of 17.5” for the

factor.

**Definition**: Let the symbol, *n*_{c}, refer to the number of center points in a central composite experimental design with the so-called block factor having a value of 1.

**Origin of RSM Designs and Decision-making:**

In this section, the origins of the experimental planning matrices used in standard responses surface methods are described. The phrase “**experimental arrays**” is used to describe the relevant planning matrices. Also, information that can aid in decision-making about which array should be used is provided.

**Origins of the RSM Experimental Arrays:**

A. Box and Wilson (1951) generated CCD arrays by combining three components as indicated by the example. For clarity, lists the design in standard order (SO), which is not randomized. To achieve proof and avoid problems, the matrix should not be used in this order. The run order should be randomized.

B. The first CCD component consists of a two level matrix similar or identical to the ones used for screening Specifically, this portion is either a full factorials or a so-called **“Resolution V” **regular fractional factorial.

C. The phrase **“Resolution V”** refers to regular fractional factorials with the property that no column can be derived without multiplying at least four other columns together.

D. For example, it can be checked that a 16 run regular fractional factorial with five factors and the generator E = ABCD is Resolution V. Resolution V implies that a model form with all two level interactions, *e.g.*, β 10*x*_{2}*x*_{3} , can be fitted with accuracy that is often acceptable.

E. The phrase **“center points”** refers to experimental runs with all setting set to levels at the midpoint of the factor range. The second CCD component part consists of *nc *center points.

F. Advanced readers may realize that the quantity *s *÷ *c*4 is an “assumption-free” estimate of the random error standard deviation, σ0.

G. The quantity *s *÷ *c*_{4} only reflects random or experimental errors and is not affected by the choice of fit model form. The phrase “star points” refers to experimental runs in which a single factor is set to αC or –αC while the other factors are set at the mid values.

H. The last CCD component part consists of two star points for every factor. One desirable feature of CCDs is that the value of αC can be adjusted by the method user.

I. The statistical properties of the CCD based RSM method are often considered acceptable for 0.5 < α_{C }< sqrt[*m*], where *m *is the number of factors.