RELIABILITY IMPROVEMENT THROUGH ROBUST DESIGN
5.8 DESIGN OF EXPERIMENTS
In earlier sections we defined the scope of robust design, identified the critical control factors and noise factors and their levels, and determined the key quality characteristic. The next step in robust design is to design the experiment.
Design of experiment is a statistical technique for studying the effects of multiple factors on the experimental response simultaneously and economically.
The factors are laid out in a structured array in which each row represents a combination of levels of factors. Then experiments with each combination are conducted and response data are collected. Through experimental data analysis, we can choose the optimal levels of control factors that minimize the sensitivity of the response to noise factors.
Various types of structured arrays or experimental designs, such as full factorial designs and a variety of fractional factorial designs are described in the literature (e.g., C. F. Wu and Hamada, 2000; Montgomery, 2001b). In a full factorial design, the number of runs equals the number of levels to the power of the number of factors. For example, a two-level full factorial design with eight factors requires 28= 256 runs. If the number of factors is large, the experiments will be unaffordable in terms of time and cost. In these situations, a fractional factorial design is often employed. A fractional factorial design is a subset of a full factorial design, chosen according to certain criteria. The commonly used classical fractional factorials are 2k−p and 3k−p, where 2 (3) is the number of levels, k
DESIGN OF EXPERIMENTS 137
the number of factors, and 2−p (3−p) the fraction. For example, a two-level half-fractional factorial with eight factors needs only 28−1= 128 runs.
The classical fractional factorial designs require that all factors have an equal number of levels. For example, a 2k−p design can accommodate only two-level factors. In practice, however, some factors are frequently required to take a differ-ent number of levels. In such situations, the classical fractional factorial designs are unable to meet the demand. A more flexible design is that of orthogonal arrays, which have been used widely in robust design. As will be shown later, the classical fractional factorial designs are special cases of orthogonal arrays. In this section we present experimental design using orthogonal arrays.
5.8.1 Structure of Orthogonal Arrays
An orthogonal array is a balanced fractional factorial matrix in which each row represents the levels of factors of each run and each column represents the levels of a specific factor that can be changed from each run. In a balanced matrix:
ž All possible combinations of any two columns of the matrix occur an equal number of times within the two columns. The two columns are also said to be orthogonal.
ž Each level of a specific factor within a column has an equal number of occurrences within the column.
For example, Table 5.1 shows the orthogonal array L8(27). The orthogonal array has seven columns. Each column may accommodate one factor with two levels, where the low and high levels are denoted by 0 and 1, respectively. From Table 5.1 we see that any two columns, for example, columns 1 and 2, have level combinations (0,0), (0,1), (1,0), and (1,1). Each combination occurs twice within the two columns. Therefore, any two columns are said to be orthogonal.
In addition, levels 0 and 1 in any column repeat four times. The array contains eight rows, each representing a run. A full factorial design with seven factors and
TABLE 5.1 L8(27) Orthogonal Array Column
Run 1 2 3 4 5 6 7
1 0 0 0 0 0 0 0
2 0 0 0 1 1 1 1
3 0 1 1 0 0 1 1
4 0 1 1 1 1 0 0
5 1 0 1 0 1 0 1
6 1 0 1 1 0 1 0
7 1 1 0 0 1 1 0
8 1 1 0 1 0 0 1
two levels of each would require 27= 128 runs. Thus, this orthogonal array is a 161 fractional factorial design. In general, because of the reduction in run size, an orthogonal array usually saves a considerable amount of test resource. As opposed to the improved test efficiency, an orthogonal array may confound the main effects (factors) with interactions. To avoid or minimize such confounding, we should identify any interactions before design of experiment and lay out the experiment appropriately. This is discussed further in subsequent sections.
In general, an orthogonal array is indicated by LN(IP× JQ), where N denotes the number of experimental runs, P is the number of I-level columns, and Q is the number of J-level columns. For example, L18(21× 37) identifies the array as having 18 runs, one two-level column, and seven three-level columns. The most commonly used orthogonal arrays have the same number of levels in all columns, and then LN(IP× JQ) simplifies to LN(IP). For instance, L8(27) indicates that the orthogonal array has seven columns, each with two levels. The array requires eight runs, as shown in Table 5.1.
Because of the orthogonality, some columns in LN(IP) are fundamental (inde-pendent) columns, and all other columns are generated from two or more of the fundamental columns. The generation formula is as follows, with few exceptions.
(number in the column generated from ifundamental columns)
=
i j=1
(number in fundamental columnj )(mode I), (5.12)
where 2≤ i ≤ total number of fundamental columns. The modulus I calculation gives the remainder after the sum is divided by I.
Example 5.2 In L8(27) as shown in Table 5.1, columns 1, 2, and 4 are the fun-damental columns, all other columns being generated from these three columns.
For instance, column 3 is generated from columns 1 and 2 as follows:
column 1
As explained in Section 5.8.1, an orthogonal array is comprised of fundamen-tal columns and generated columns. Fundamenfundamen-tal columns are the independent
DESIGN OF EXPERIMENTS 139
columns, and generated columns are the interaction columns. For example, in Table 5.1, the interaction between columns 1 and 2 goes to column 3. In an experimental layout, if factor A is assigned to column 1 and factor B to column 2, column 3 should be allocated to the interaction A× B if it exists. Assigning an independent factor to column 3 can lead to an incorrect data analysis and faulty conclusion because the effect of the independent factor is confounded with the interaction effect. Such experimental design errors can be prevented by using a linear graph.
A linear graph, a pictorial representation of the interaction information, is made up of dots and lines. Each dot indicates a column to which a factor (main effect) can be assigned. The line connecting two dots represents the interaction between the two factors represented by the dots at each end of the line segment.
The number assigned to a dot or a line segment indicates the column within the array. In experimental design, a factor is assigned to a dot, and an interaction is assigned to a line. If the interaction represented by a line is negligible, a factor may be assigned to the line.
Figure 5.12 shows two linear graphs of L8(27). Figure 5.12a indicates that columns 1, 2, 4, and 7 can be used to accommodate factors. The interaction between columns 1 and 2 goes into column 3, the interaction between 2 and 4 goes into column 6, and the interaction between columns 1 and 4 goes into column 5. From (5.12) we can see that column 7 represents a three-way interaction among columns 1, 2, and 4. The linear graph assumes that three-way or higher-order interactions are negligible. Therefore, column 7 is assignable to a factor. It should be noted that all linear graphs are based on this assumption, although it may be questionable in some applications.
Example 5.3 To assign an experiment with five two-level factors, A, B, C, D, and E, and interactions A× B and B × C to L8(27), using Figure 5.12a we can allocate factor A to column 1, factor B to column 2, factor C to column 4, factor D to column 7, factor E to column 5, interaction A× B to column 3, and interaction B× C to column 6.
Most orthogonal arrays have two or more linear graphs. The number and complexity of linear graphs increase with the size of orthogonal array. A multitude of linear graphs provide great flexibility for assigning factors and interactions.
(a) (b)
5 3
6
2
4 7 1
1
2 4
3 5 7
6
FIGURE 5.12 Linear graphs for L8(27)
The most commonly used orthogonal arrays and their linear graphs are listed in the Appendix.
5.8.3 Two-Level Orthogonal Arrays
A two-level orthogonal array is indicated as LN(2P). The most frequently used two-level arrays are L4(23), L8(27), L12(211), and L16(215).
L4(23), shown in Table 5.2, is a half-fractional factorial array. The first two columns can be assigned to two factors. The third column accommodates the interaction between them, as indicated by the linear graph in Figure 5.13. If a negligible interaction can be justified, the third column is assignable to an additional factor.
The layout and linear graphs of L8(27) are shown in Table 5.1 and Figure 5.12, respectively. This array requires only eight runs and is very flexible in investi-gating the effects of factors and their interactions. It is often used in small-scale experimental designs.
L12(211), given in the Appendix, is unique in that the interaction between any two columns within the array is partially spread across the remaining columns.
This property minimizes the potential of heavily confounding the effects of factors and interactions. If engineering judgment considers the interactions to be weak, the array is efficient in investigating the main effects. However, the array cannot be used if the interactions must be estimated.
L16(215) is often used in a large-scale experimental design. The array and the commonly used linear graphs can be found in the Appendix. This array provides great flexibility for examining both simple and complicated two-way interactions.
For example, it can accommodate 10 factors and five interactions, or five factors and 10 interactions.
5.8.4 Three-Level Orthogonal Arrays
Three-level orthogonal arrays are needed when experimenters want to explore the quadratic relationship between a factor and the response. Although the response
TABLE 5.2 L4(23) Orthogonal Array Column
Run 1 2 3
1 0 0 0
2 0 1 1
3 1 0 1
4 1 1 0
1 3 2
FIGURE 5.13 Linear graph for L4(23)
DESIGN OF EXPERIMENTS 141
surface method is deemed advantageous in investigating such a relationship, experimental design using orthogonal arrays is still widely used in practice because of the simplicity in data analysis. Montgomery (2001b) describes response surface analysis.
LN(3P) refers to a three-level orthogonal array where 0, 1, and 2 denote the low, middle, and high levels, respectively. Any arrays in which the columns contain predominantly three levels are also called three-level orthogonal arrays.
For example, L18(21× 37) is considered a three-level array. The most commonly used three-level arrays are L9(34), L18(21× 37), and L27(313).
L9(34) is the simplest three-level orthogonal array. As shown in Table 5.3, this array requires nine runs and has four columns. The first two columns are the fundamental columns, and the last two accommodate the interactions between columns 1 and 2, as shown in Figure 5.14. If no interactions can be justified, the array can accommodate four factors.
L18(21× 37) is a unique orthogonal array (see the Appendix for the layout and linear graphs of this array). The first column in this array contains two levels and all others have three levels. The interaction between the first two columns is orthogonal to all columns. Therefore, the interaction can be estimated without sacrificing additional columns. However, the interactions between any pair of three-level columns are spread to all other three-level columns. If the interactions between three-level factors are strong, the array cannot be used.
L27(313) contains three levels in each of its 13 columns (the layout and linear graphs are given in the Appendix). The array can accommodate four interactions and five factors, or three interactions and seven factors. Because the interaction between two columns spreads to the other two columns, two columns must be sacrificed in considering one interaction.
TABLE 5.3 L9(34) Orthogonal Array Column
Run 1 2 3 4
1 0 0 0 0
2 0 1 1 1
3 0 2 2 2
4 1 0 1 1
5 1 1 2 0
6 1 2 0 1
7 2 0 2 1
8 2 1 0 2
9 2 2 1 0
1 2
3, 4
FIGURE 5.14 Linear graph for L9(34)
5.8.5 Mixed-Level Orthogonal Arrays
The orthogonal arrays that have been discussed so far can accommodate only two-or three-level facttwo-ors. Except ftwo-or L18(21× 37), the arrays require that all factors have an equal number of levels. In practice, however, we frequently encounter situations in which few factors have more levels than all others. In such cases, the few factors with multiple levels will considerably increase the array size. For example, L8(27) is good for investigating five independent factors. However, if one of the five factors has to take four levels, L16(45) is needed simply because of the four-level factor, apparently rendering the experiment not economically efficient. To achieve the economic efficiency, in this section we describe the preparation of mixed-level orthogonal arrays using the column-merging method.
The method is based on a linear graph and the concept of degrees of freedom (Section 5.8.6). In particular, in this section we describe a method of preparing a four-level column and an eight-level column in standard orthogonal arrays.
First, we study a method of creating a four-level column in a standard two-level orthogonal array. Because a four-two-level column has three degrees of freedom and a two-level column has one, the formation of one four-level column requires three two-level columns. The procedure of forming a four-level column has three steps:
1. Select any two independent (fundamental) columns and their interaction column. For example, to generate a four-level column in L8(27), columns 1, 2, and 3 may be selected, as shown in Figure 5.15.
2. Merge the numbers of the two independent (fundamental) columns selected and obtain 00, 01, 10, and 11, denoted 0, 1, 2, and 3, respectively. Then the merging forms a new column whose levels are 0, 1, 2, and 3. In the L8(27) example, combining the numbers of columns 1 and 2 gives a new series of numbers, as shown in Table 5.4.
3. Replace the three columns selected with the four-level column. In the L8(27) example, the first three columns are replaced by the new column. The new column is orthogonal to any other column, except for the original first three columns. Now, a four-level factor can be assigned to the new column, and other two-level factors go to columns 4, 5, 6, and 7.
In experimentation, an eight-level column is sometimes needed. As with a four-level column, an eight-four-level column can be prepared using the column-merging
5 3
6
2
4
7 1
FIGURE 5.15 Selection of three columns to form a new column
DESIGN OF EXPERIMENTS 143
TABLE 5.4 Formation of a Four-Level Column in L8(27)
FIGURE 5.16 Selection of seven columns to form a new column
method. Because an eight-level column counts for seven degrees of freedom, it can be obtained by combining seven two-level columns. The procedure is similar to that for creating a four-level column.
1. Select any three independent (fundamental) columns and their four inter-action columns with the assistance of linear graphs. For example, if an eight-level column is to be created in L16(215), we can choose the indepen-dent columns 1, 2, and 4 and their interaction columns, 3, 5, 6 and 7, as shown in Figure 5.16. Column 7 accommodates three-column interaction (columns 1, 2, and 4) or two-column interaction (columns 1 and 6).
2. Merge the numbers of the three independent (fundamental) columns sele-cted and obtain 000, 001, 010, 011, 100, 101, 110, and 111, denoted by 0, 1, 2, 3, 4, 5, 6, and 7, respectively. Then the merging forms a new column with these eight levels. In the L16(215) example, combining the numbers of columns 1, 2, and 4 gives a new series of numbers, as shown in Table 5.5.
3. Replace the seven columns selected with the eight-level column. In the L16(215) example, columns 1 to 7 are replaced by the new column. The new column is orthogonal to any other column, except for the original seven columns. Now, the new array can accommodate an eight-level factor and up to eight two-level factors.
TABLE 5.5 Formation of an Eight-Level Column in L16(215) Column
Run 1 2 4
Combined Number
New
Column 8 9 10 11 12 13 14 15
1 0 0 0 000 0 0 0 0 0 0 0 0 0
2 0 0 0 000 0 1 1 1 1 1 1 1 1
3 0 0 1 001 1 0 0 0 0 1 1 1 1
4 0 0 1 001 1 1 1 1 1 0 0 0 0
5 0 1 0 010 2 0 0 1 1 0 0 1 1
6 0 1 0 010 2 1 1 0 0 1 1 0 0
7 0 1 1 011 3 0 0 1 1 1 1 0 0
8 0 1 1 011 3 1 1 0 0 0 0 1 1
9 1 0 0 100 4 0 1 0 1 0 1 0 1
10 1 0 0 100 4 1 0 1 0 1 0 1 0
11 1 0 1 101 5 0 1 0 1 1 0 1 0
12 1 0 1 101 5 1 0 1 0 0 1 0 1
13 1 1 0 110 6 0 1 1 0 0 1 1 0
14 1 1 0 110 6 1 0 0 1 1 0 0 1
15 1 1 1 111 7 0 1 1 0 1 0 0 1
16 1 1 1 111 7 1 0 0 1 0 1 1 0
5.8.6 Assigning Factors to Columns
To lay out an experiment, we must select an appropriate orthogonal array and allocate the factors and interactions to the columns within the array. The selection of an orthogonal array is directed by the concept of degrees of freedom, and the assignment of factors and interactions is assisted by linear graphs.
The term degrees of freedom has different meanings in physics, chemistry, engineering, and statistics. In statistical analysis, it is the minimum number of comparisons that need to be made to draw a conclusion. For example, a factor of four levels, say A0, A1, A2, and A3, has three degrees of freedom because we need three comparisons between A0 and the other three levels to derive a conclusion concerning A0. Generally, in the context of experimental design, the number of degrees of freedom required to study a factor equals the number of factor levels minus one. For example, a two-level factor counts for 1 degree of freedom, and a three-level factor has 2 degrees of freedom.
The number of degrees of freedom of an interaction between factors equals the product of the degrees of freedom of the factors comprising the interaction.
For example, the interaction between a three-level factor and a four-level factor has (3− 1) × (4 − 1) = 6 degrees of freedom.
The number of degrees of freedom in an orthogonal array equals the sum of degrees of freedom available in each column. If we continue to use LN(IP) to denote an orthogonal array, the degrees of freedom of the array would be (I− 1) × P. For example, the number of degrees of freedom available in L16(215) is(2− 1) × 15 = 15. L18(21× 37) is a special case that deserves more attention.
DESIGN OF EXPERIMENTS 145
As shown in Section 5.8.4, interaction between columns 1 and 2 of the array is orthogonal to all other columns. The interaction provides(2− 1) × (3 − 1) = 2 degrees of freedom. Then the total number of degrees of freedom is 2+ (2 − 1)× 1 + (3 − 1) × 7 = 17. From the examples we can see that the number of degrees of freedom in an array equals the number of runs of the array minus one, that is,N− 1. This is generally true because N runs of an array provide N − 1 degrees of freedom.
Having understood the concept and calculation of degrees of freedom, we can select an appropriate orthogonal array and assign factors and interactions to the columns in the array by using its linear graphs. The procedure is as follows:
1. Calculate the total number of degrees of freedom needed to study the factors (main effects) and interactions of interest. This is the degrees of freedom required.
2. Select the smallest orthogonal array with at least as many degrees of free-dom as required.
3. If necessary, modify the orthogonal array by merging columns or using other techniques to accommodate the factor levels.
4. Construct a required linear graph to represent the factors and interactions.
The dots represent the factors, and the connecting lines indicate the inter-actions between the factors represented by the dots.
5. Choose the standard linear graph that most resembles the linear graph required.
6. Modify the required graph so that it is a subset of the standard linear graph.
7. Assign factors and interactions to the columns according to the linear graph.
The unoccupied columns are error columns.
Example 5.4 The rear spade in an automobile can fracture in the ends of the structure due to fatigue under road conditions. An experiment was designed to improve the fatigue life of the structure. The fatigue life may be affected by the setting of the design parameters as well as the manufacturing process.
The microfractures generated during forging grow while the spade is in use.
Therefore, the control factors in this study include the design and production process parameters. The main control factors are as follows:
ž Factor A: material; A0= type 1, A1 = type 2
ž Factor B: forging thickness; B0= 7.5 mm, B1= 9.5 mm
ž Factor C: shot peening; C0= normal, C1= masked
ž Factor D: bend radius; D0= 5 mm, D1= 9 mm
In addition to these main effects (factors), interactions B× D and C × D are possible and should be included in the study. Select an appropriate orthogonal array and lay out the experiment.
D
B
C× D C
B× D
A
(a)
1
2 4
3 5
7
6 (b)
FIGURE 5.17 (a) Required linear graph; (b) standard linear graph
1 D
2 4
3 5
7 A
error column
B C
B× D C× D
6
FIGURE 5.18 Assignment of factors and interactions to L8(27)
SOLUTION To design the experiment, we first calculate the degrees of freedom for the factors and interactions. Each factor has two levels and thus has 2− 1 = 1 degree of freedom. Each interaction has(2− 1) × (2 − 1) = 1 degree of freedom.
The total number of degrees of freedom equals 4× 1 + 2 × 1 = 6. Then we select L8(27), which provides seven degrees of freedom, to lay out the experiment.
The total number of degrees of freedom equals 4× 1 + 2 × 1 = 6. Then we select L8(27), which provides seven degrees of freedom, to lay out the experiment.