3. Six Sigma Experiences and Leadership
4.6 Design of Experiments (DOE)
Figure 4.16. Outline of experimental design procedure 2. The development of new processes. The application of
DOE methods early in process development can result in reduced development time, reduced variability of target requirements, and enhanced process yields.
3. Screening important factors.
4. Engineering design activities such as evaluation of mate- rial alternations, comparison of basic design configura- tions, and selection of design parameters so that the product is robust to a wide variety of field conditions.
5. Empirical model building to determine the functional relationship between x and y.
The tool, DOE, was developed in the 1920s by the British scientist Sir Ronald A. Fisher (1890–1962) as a tool in agricul- tural research. The first industrial application was performed in order to examine factors leading to improved barley growth for the Dublin Brewery. After its original introduction to the brewery industry, factorial design, a class of design in DOE, began to be applied in industries such as agriculture, cotton, wool and chemistry. George E. P. Box (1919–), an American
Statement of the experimental problem
Understanding of present situation
Choice of response variables
Confirmation test
Analysis of results and conclusions
Data analysis Choice of factors
and levels analysis
Selection of experimental design
Performing the experiments Planning of subsequent
experiments
Recommendation and follow-up management
scientist, and Genichi Taguchi (1924–), a Japanese scientist, have contributed significantly to the usage of DOE where vari- ation and design are the central considerations.
Large manufacturing industries in Japan, Europe and the US have applied DOE from the 1970s. However, DOE remained a specialist tool and it was first with Six Sigma that DOE was brought to the attention of top management as a powerful tool to achieve cost savings and income growth through improvements in variation, cycle time, yield, and design. DOE was also moved from the office of specialists to the corporate masses through the Six Sigma training scheme.
(2) Classification of design of experiments
There are many different types of DOE. They may be clas- sified as follows according to the allocation of factor combi- nations and the degree of randomization of experiments.
1. Factorial design: This is a design for investigating all possi- ble treatment combinations which are formed from the fac- tors under consideration. The order in which possible treat- ment combinations are selected is completely random. Sin- gle-factor, two-factor and three-factor factorial designs belong to this class, as do 2k(k factors at two levels) and 3k (k factors at three levels) factorial designs.
2. Fractional factorial design: This is a design for investigating a fraction of all possible treatment combinations which are formed from the factors under investigation. Designs using tables of orthogonal arrays, Plackett-Burman designs and Latin square designs are fractional factorial designs. This type of design is used when the cost of the experiment is high and the experiment is time-consuming.
3. Randomized complete block design, split-plot design and nested design: All possible treatment combinations are test- ed in these designs, but some form of restriction is imposed
on randomization. For instance, a design in which each block contains all possible treatments, and the only ran- domization of treatments is within the blocks, is called the randomized complete block design.
4.Incomplete block design: If every treatment is not present in every block in a randomized complete block design, it is an incomplete block design. This design is used when we may not be able to run all the treatments in each block because of a shortage of experimental apparatus or inadequate facilities.
5. Response surface design and mixture design: This is a design where the objective is to explore a regression model to find a functional relationship between the response variable and the factors involved, and to find the optimal conditions of the factors. Central composite designs, rotatable designs, simplex designs, mixture designs and evolutionary opera- tion (EVOP) designs belong to this class. Mixture designs are used for experiments in which the various components are mixed in proportions constrained to sum to unity.
6. Robust design: Taguchi (1986) developed the foundations of robust design, which are often called parameter design and tolerance design. The concept of robust design is used to find a set of conditions for design variables which are robust to noise, and to achieve the smallest variation in a product’s function about a desired target value. Tables of orthogonal arrays are extensively used for robust design.
For references related to robust design, see Taguchi (1987), Park (1996) and Logothetis and Wynn (1989).
(3) Example of 23factorial design
There are many different designs that are used in industry.
A typical example is illustrated here. Suppose that three fac- tors, A, B and C, each at two levels, are of interest. The design
is called a 23factorial design, and the eight treatment combi- nations are written in Table 4.5 and they can be displayed graphically as a cube, as shown in Figure 4.17. We usually write the treatment combinations in standard order as (1), c, b, bc, a, ac, ab, abc.
There are actually three different notations that are widely used for the runs in the 2k design. The first is the “+ and –” nota- tion, and the second is the use of lowercase letters to identify the treatment combinations. The final notation uses 1 and 0 to denote high and low factor levels, respectively, instead of + and 1.
Table 4.5. 23runs and treatment combinations
Figure 4.17. 23factorial design
y8(abc) y4(bc)
y6(ac) y2(c)
y8(abc) y4(bc)
y6(ac) y2(c)
(+)
(–) C
(+)
(–) B
(+) (–)
A
2.0 1.0
–1.4 –1.0
4.0 3.5
–2.6 –2.5
(+)
(–) C
(+)
(–) B
(+) (–)
A
A B C A B C
Run (+/– notation)
Treatment
combinations (1/0 notation) Response data 1
2 3 4 5 6 7 8
– – – – + + + +
– – + + – – + +
– + – + – + – +
(1) c b bc
a ac ab abc
0 0 0 0 1 1 1 1
0 0 1 1 0 0 1 1
0 1 0 1 0 1 0 1
– 2.5 – 1.0 3.5 1.0 – 2.6 – 1.4 4.0 2.0
=3.0
= yi
T Σ
Suppose that a soft drink bottler is interested in obtaining more uniform fill heights in the bottles produced by his man- ufacturing process. The filling machine theoretically fills each bottle to the correct target height, but in practice, there is vari- ation around this target, and the bottler would like to under- stand the sources of this variability and eventually reduce it.
The process engineer can control three variables during the filling process as given below, and the two levels of experi- mental interest for each factor are as follows:
A: The percentage of carbonation (A0= 10%, A1= 12%) B: The operating pressure in the filler (B0 = 25 psi, B1 =
30 psi)
C: The line speed (C0= 200 bpm, C1= 250 bpm)
The response variable observed is the average deviation from the target fill height observed in a production run of bot- tles at each set of conditions. The data that resulted from this experiment are shown in Table 4.5. Positive deviations are fill heights above the target, whereas negative deviations are fill heights below the target.
The analysis of variance can be done as follows. Here, Tiis the sum of four observations at the level of Ai, and Tijis the sum of two observations at the joint levels of AiBj. The ANOVA (analysis of variance) table can be summarized as shown in Table 4.6.
squares of
sum corrected total
T = S
8 )
( 2
2 −
= i i
y y
095 . 8 48
) 0 . 3 ) ( 0 . 2 ( )
0 . 1 ( ) 5 . 2 (
2 2
2
2 + − + + − =
−
= ... .
Σ Σ
Similarly, we can find that SB= 40.5, SC= 0.405. For the inter- action sum of squares, we can show that
Similarly, we can find that SA×C= 0.005 and SB×C= 6.48. The error sum of squares can be calculated as
Table 4.6. ANOVA table for soft drink bottling problem
Source of variation Sum of squares Degrees of freedom Mean square F0
A B C A×B A×C B×C Error(e)
0.125 40.500 0.405 0.500 0.005 6.480 0.080
1 1 1 1 1 1 1
0.125 40.500 0.405 0.500 0.005 6.480 0.080
1.56 506.25 5.06 6.25 0.06 81.00
Total 48.095 7
08 . 0 )
( =
− + + + + +
= T A B C A×B A×C B×C
e S S S S S S S
S .
2 10•
01•
00•
8 11•
1 + − −
= T T T T
SA×B
2
8
1 ab + abc + (1) + c – b – bc – a – ac
=
2
8
1 4.0+2.0+(–2.5)+(–1.0)–3.5–1.0–(–2.6)–(–1.4)
= 5 .
=0 .
2 10•
01•
00•
8 11•
1 + − −
= T T T T
SA
2
8
1 ab + abc + (1) + c – b – bc – a – ac
=
2
8
1 4.0+2.0+(–2.5)+(–1.0)–3.5–1.0–(–2.6)–(–1.4)
= 5 .
=0 .
Since the F0 value of A×C is less than 1, we pool A×C into the error term, and the pooled ANOVA table can be constructed as follows.
Table 4.7. Pooled ANOVA table for soft drink bottling problem
To assist in the practical interpretation of this experiment, Figure 4.18 presents plots of the three main effects and the A×B and B×C interactions. Since A×C is pooled, it is not plot- ted. The main effect plots are just graphs of the marginal response averages at the levels of the three factors. The inter- action graph of A×B is the plot of the averages of two responses at A0B0, A0B1, A1B0 and A1B1. The interaction graph of B×C can be similarly sketched. The averages are shown in Table 4.8.
Table 4.8. Averages for main effects and interactions
A0 A1 B0 B1 C0 C1
0.25 0.50 –1.875 2.625 0.6 0.15
A0 A1 B0 B1
B0 –1.75 –2.0 C0 –2.55 3.75
B1 2.25 3.0 C1 –1.2 1.5
Source of variation Sum of squares Degrees of freedom Mean square F0
A B C A×B B×C Pooled error(e)
0.125 40.500 0.405 0.500 6.480 0.085
1 1 1 1 1 2
0.125 40.500 0.405 0.500 6.480 0.0425
2.94 952.94**
9.53∆ 11.76∆ 152.47**
Total 48.095 7
** : Significant at 1% level.
∆ : Significant at 10% level.
Figure 4.18. Main effects and interaction plots
Notice that two factors, A and B, have positive effects; that is, increasing the factor level moves the average deviation from the fill target upward. However, factor C has a negative effect. The interaction between B and C is very large, but the interaction between A and B is fairly small. Since the compa- ny wants the average deviation from the fill target to be close to zero, the engineer decides to recommend A0B0C1 as the optimal operating condition from the plots in Figure 4.18.