## 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.

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 2^{7} = 128 experiments. Taguchi's L_{8} 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, A_{1} and A_{2}); and for Level 3 factors, the levels are denoted 1, 2, and 3 (or for A_{1}, A_{2}, and A_{3}).

As Figure 1-11 shows, an L_{8} 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 L_{12} array, which is described in Chapter 2. As discussed there, use of L_{12}, L_{18}, L_{36}, or L_{54} 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.