Example of a 23 Factorial Experiment

Below is a hypothetical example of a 23 factorial experiment to illustrate the application of factorial experiments in improving processes.

In this experiment, the process engineer's goal is to determine how the yield of an adhesive application process can be improved by adjusting three (3) process parameters:  mixture ratio, curing temperature, and curing time.  For each of these input parameters, two levels will be defined for use in this 2-level experiment.  For the mix ratio, the high level is set at 55%, while the low level is set at 45%.  For the curing temp., the high level is set at 150 deg C while the low level is set at 100 deg C.  For the curing time, the high level is set at 90 minutes, while the low level is set at 30 minutes. As mentioned, the output response monitored is process yield. Assume further that the data were gathered by performing just a single replicate (n=1) per combination treatment.

Table 1. Results of the Example 23 Factorial Experiment

 RUN Comb. Factors Yield Mix Ratio Temp Time 1 (1) 45% (-) 100C (-) 30m (-) 8 2 a 55% (+) 100C (-) 30m (-) 9 3 b 45% (-) 150C (+) 30m (-) 34 4 ab 55% (+) 150C (+) 30m (-) 52 5 c 45% (-) 100C (-) 90m (+) 16 6 ac 55% (+) 100C (-) 90m (+) 22 7 bc 45% (-) 150C (+) 90m (+) 45 8 abc 55% (+) 150C (+) 90m (+) 56

Applying Table 6 of the article factorial design tables to get the algebraic signs of the coefficients of the factorial effect formulas as discussed in the article on 2-Level factorial experiments, the following calculations for the main and interaction effects of these 3 factors are obtained:

A = 1/(4n) x [-(1)+a-b+ab-c+ac-bc+abc] = [-8+9-34+52-16+22-45+56] = 1/4 x 36 = 9

B = 1/4 x [-8-9+34+52-16-22+45+56] = 1/4 x 132 = 33

AB = 1/4 x [+8-9-34+52+16-22-45+56] = 1/4 x 22 = 5.5

C = 1/4 x [-8-9-34-52+16+22+45+56] = 1/4 x 36 = 9

AC = 1/4 x [+8-9+34-52-16+22-45+56] = 1/4 x  -2 = -0.5

BC = 1/4 x [+8+9-34-52-16-22+45+56] = 1/4 x -6 = -1.5

ABC = 1/4 x [-8+9+34-52+16-22-45+56] = 1/4 x -12 = -3

Based on these calculations, the main effect of temperature (B=33) has the greatest influence on the process yield, although the main effects of mixture ratio (A=9) and time (C=9) are also significant.  The interaction between mixture ratio and temperature also produces a positive effect on yield (AB=5.5), but the rest of the factorial interactions affect the yield in the negative direction (although to much lower degrees).