This chapter is from the book
This chapter is from the book
This chapter is from the book
1.7 Assign Control Factors to the Inner Array
The sixth step in the robust design process is to assign control factors to an inner array. Orthogonal arrays are efficient tools for multifactor experimentation. The orthogonal array of control factors is called the inner array. The inner array specifies the combinations of control factor levels to be tested. When running a designed experiment, the present design (or other reference design) should be included, allowing comparison of the current process to alternatives based on common testing conditions.
Figure 1-9 shows a sample orthogonal array. Genichi Taguchi (1992, 1999, 2000) made a significant contribution by adapting fractional factorial orthogonal arrays (balanced both ways) to experimental design so that time and cost of experimentation are reduced while validity and reproducibility are maintained.
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Figure 1-9 An orthogonal array.
Taguchi's approach is disciplined and structured to make it easy for quality engineers to apply. Use of orthogonal arrays has been demonstrated to produce efficient robust designs that improve product development productivity. A full factorial design with seven factors at two levels would require 27 = 128 experiments. Taguchi's L8 orthogonal array requires only eight experiments. Typically, orthogonal arrays include a configuration in which all factors are set at Level 1. When using these arrays, the team may elect to let Level 1 represent the present design so that no testing beyond that specified in the inner array is necessary. Other teams may prefer to assign levels in increasing order of the factor settings so that it is easy to interpret the response tables and plots relative to the settings.
When testing at only two levels, the team may opt to test at levels above and below the present level. If this is preferred, the reference design should be run in addition to those specified in the inner array. Although the reference design statistics will be used for comparison to the selected optimal, they should not be included when developing response tables and plots.
Based on their relative impact on the system and on available resources, the team must now select the control factors for experimentation. They should then identify each factor's level and assign the factors and levels to the inner array. Previous experience, studies, or screening experiments can be used to help prioritize the brainstormed list of control factors.
Parameter Design experiments should be conducted with low-cost alternatives to present design settings (see Chapter 7). (Higher-cost alternatives are considered in Chapter 8, Tolerance Design, which emphasizes cost and quality tradeoffs.) As many factors as possible should be identified to enhance improvement potential. Control factors are usually tested at two or three levels in orthogonal array experiments; however, techniques are available to accommodate more levels.
The range of levels should be broad but still maintain system function. If the system ceases to function at a combination of factor levels designated by the inner array, data will be unavailable for a run. As a result, balance will be lost and all affect estimates will be biased.
In this book, control factor levels are denoted with numerals. Thus, for Level 2 factors, the levels are denoted 1 and 2 (or for Factor A, A1 and A2); and for Level 3 factors, the levels are denoted 1, 2, and 3 (or for A1, A2, and A3).
Example 1.2
Let's say you are an engineer working on the ball-swirling line at the Marion Bearing Manufacturing plant. As a cost-saving measure, management would like to loosen the ball bearing diameter's tolerance. As shown in Figure 1-10, the swirl-time variation has a significant impact on bearing quality.
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Figure 1-10 Quality problems caused by variations in swirl time.
The Six Sigma project champion has assigned your team the task of establishing process parameter nominal values at which the swirl time in the swirling machine's funnel will be robust to variation in ball bearing diameter in order to achieve a target in 10 seconds. After lab or project group assignments have been made, your team will be given an experimental apparatus with which to take data.
For each run, one group member will release the ball, another will time the ball with a stopwatch, and a third will record the result on paper. The experimental apparatus consists of a funnel mounted on a stand, a ramp mounted above the funnel, and a ball bearing. Your task in this lab is to conduct an experiment to determine how long it takes the ball bearing to roll down the funnel. Teams will perform one set of three runs with each possible pairing of group members. In other words, if there are four in your group, 12 sets of 3 runs each will be done.
A data sheet on which to record the observations is provided. For each run, you record the set number, the name of the persons releasing and timing, the order of the run in the set (1 to 3), and the time. Table 1-1 shows the control factors and levels that are most likely to impact energy transfer in the ball-swirling process. Levels are the different settings a factor can have. For example, if you want to determine how the response (swirl time) is affected by the factor (run length), you would need to set the factor at different levels (e.g., 900mm and 600mm).
Table 1-1. Control Factors for Ball-Swirling Line
|
Control Factors |
Level 1 |
Level 2 |
|
L: Run length |
900mm |
600mm |
|
A: Ramp-to-funnel angle |
30 degrees |
45 degrees |
|
H: Run end height |
500mm |
600mm |
|
C: Clamping (unscrewed) |
0.0 turns |
0.5 turns |
|
O: Operator training |
Yes |
No |
For maximal test efficiency, this particular team elected to use the L8 array for the inner array of their experimental plan. As shown in Figure 1-11, the ramp-to-funnel angle was assigned to column 1 to limit the number of changes necessary. It was believed that operator training would not interact with any of the other factors, so it was assigned to column 3, which then keeps the other four main effects free from confounding by any potentially real control-by-control interactions.
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Figure 1-11 Use of an L8 orthogonal array for a swirling machine's robust design.
As Figure 1-11 shows, an L8 orthogonal array enables selection of up to seven factors for testing with only eight runs. In comparison, a 2 K full factorial design of experiment (DOE) [1] requires 128 runs. A full factorial DOE measures the response of every possible combination of factors and factor levels. These responses are analyzed to provide information about every main effect and every interaction effect.
A full factorial DOE is practical when fewer than five factors are being investigated. Testing all combinations of factor levels becomes too expensive and time-consuming with five or more factors. Orthogonal arrays include selected combinations of factors and levels. It is a carefully prescribed and representative subset of a full factorial design. By reducing the number of runs, orthogonal arrays will not be able to evaluate the impact of some of the factors independently. In general, higher-order interactions are confounded with main effects or lower-order interactions. Because higher-order interactions are rare, usually the assumption is that their effect is minimal and that the observed effect is caused by the main effect or lower-level interaction.
If more than seven factors were selected for testing, it may have been more practical to use the L12 array, which is described in Chapter 2. As discussed there, use of L12, L18, L36, or L54 orthogonal arrays is recommended. These arrays allow for testing many factors and share the quality that only fractions of interaction effects confound the main effects in any column.