# Example 6 - Taguchi Orthogonal Array Design

## Background

Taguchi orthogonal array (OA) designs are often used in design experiments with multiple level factors. Taguchi OA can be thought of as a general fractional factorial design.

Consider an experiment to study the effect of four three-level factors on a fine gold wire bonding process in an IC chip-package.* A Taguchi OA L27 (3^13) design is applied to identify the critical parameters in the wire bonding process. The response is the ball size. The smaller the ball size, the better the process.

For this example, the four factors are:

 Name Level 1 Level 2 Level 3 Force 5 10 15 Power 40 50 60 Time 15 20 25 Temperature 155 160 165

## Experiment Design

To find the significant effects, the experimenters use the standard design folio in DOE++ to create a Taguchi OA. On the Design tab of the folio, the following design settings are specified. The design matrix and response data are given on the Data tab of the "Taguchi OA L27(3^13)" folio.

In addition to these settings, the four factors are assigned to columns 1, 2, 5 and 8 in the L27 array, respectively.

Figure 1: Assignment of factors to columns in the L27 array.

The remaining columns in the array either represent the interaction effects of these factors or are treated as dummy factors. More detailed discussion on the properties of Taguchi arrays can be found in the appendix of Taguchi’s handbook.**

## Analysis Part I - Identifying the Significant Effects

Step 1: When the design is built, the experiment runs are shown in a randomized order on the Data tab. After performing the runs in the displayed order and recording the results, the experimenters enter the data set into the folio.

Figure 2: The data set entered on the Data tab of the folio.

Step 2: The experimenters are interested in investigating all the main effects, as well as the interaction effects AB, AC and AD. So these terms are selected for inclusion in the analysis.

Figure 3: Select Terms window with the effects of interest selected.

Step 3: Next, the analysis settings are specified. The data set will be analyzed using the default risk (significance) level of 0.1. In addition, the test statistic for the effects will be calculated using the partial sum of squares, and the ANOVA table will show the results for each individual term.

Figure 4: Analysis settings.

Step 4: To see which effects are significant, the ANOVA table in the detailed summary of results is examined. According to the table, the main effects A, B and C are significant.

## Analysis Part II - Optimization

The results for the reduced model and the optimization are given in the "Reduced Model" folio.

Step 1: The design folio is duplicated and the copy is renamed "Reduced Model."

Step 2: Only the significant effects are selected to calculate the new model.

Figure 5: Select Terms window with only significant terms selected.

Step 3: After the reduced model is calculated, the Diagnostic Information table in the detailed summary of results is examined. According to the table, the factor settings used for the 4th experiment run (standard order = 1) will produce the smallest ball size (shown in the Fitted Value column).

 Diagnostics Standard Order Run Order Actual Value (Y) Fitted Value (YF) Residual Standardized Residual Studentized Residual External Studentized Residual Leverage Cook's Distance 5 1 38.5 39.0556 -0.5556 -0.6066 -0.7048 -0.6956 0.2593 0.0248 8 2 39 39.1111 -0.1111 -0.1213 -0.141 -0.1375 0.2593 0.001 22 3 41 40.2222 0.7778 0.8492 0.9867 0.986 0.2593 0.0487 1 4 35 35.1667 -0.1667 -0.182 -0.2114 -0.2063 0.2593 0.0022 18 5 41.5 42.4444 -0.9444 -1.0312 -1.1981 -1.2121 0.2593 0.0718 4 6 40 38.3889 1.6111 1.759 2.0438 2.2396 0.2593 0.2089 20 7 37.5 37.6667 -0.1667 -0.182 -0.2114 -0.2063 0.2593 0.0022 6 8 39.5 40.3889 -0.8889 -0.9705 -1.1276 -1.1358 0.2593 0.0636 24 9 42.5 42.2222 0.2778 0.3033 0.3524 0.3445 0.2593 0.0062 21 10 40 39 1 1.0918 1.2686 1.2894 0.2593 0.0805 2 11 36 35.8333 0.1667 0.182 0.2114 0.2063 0.2593 0.0022 15 12 42 42.3889 -0.3889 -0.4246 -0.4933 -0.4838 0.2593 0.0122 3 13 38 37.1667 0.8333 0.9098 1.0571 1.0604 0.2593 0.0559 23 14 39.5 40.8889 -1.3889 -1.5164 -1.7619 -1.8684 0.2593 0.1552 14 15 41.5 41.0556 0.4444 0.4852 0.5638 0.554 0.2593 0.0159 27 16 42 42.2778 -0.2778 -0.3033 -0.3524 -0.3445 0.2593 0.0062 7 17 38 38.4444 -0.4444 -0.4852 -0.5638 -0.554 0.2593 0.0159 11 18 38 37.8333 0.1667 0.182 0.2114 0.2063 0.2593 0.0022 25 19 41 40.2778 0.7222 0.7885 0.9162 0.9123 0.2593 0.042 26 20 42 40.9444 1.0556 1.1525 1.339 1.3679 0.2593 0.0897 13 21 40.5 40.3889 0.1111 0.1213 0.141 0.1375 0.2593 0.001 9 22 40 40.4444 -0.4444 -0.4852 -0.5638 -0.554 0.2593 0.0159 10 23 36.5 37.1667 -0.6667 -0.7279 -0.8457 -0.8394 0.2593 0.0358 19 24 35 37 -2 -2.1836 -2.5371 -3.0029 0.2593 0.3219 16 25 40.5 40.4444 0.0556 0.0607 0.0705 0.0687 0.2593 0.0002 17 26 41.5 41.1111 0.3889 0.4246 0.4933 0.4838 0.2593 0.0122 12 27 40 39.1667 0.8333 0.9098 1.0571 1.0604 0.2593 0.0559

## Conclusions

From the Fitted Value column, it is determined that run order 4 (standard order 1) gives the best result. The predicted ball size is 35.1667. The settings used for this run are Force = 5, Power = 40 and Time = 15.

In this example, all factors are assumed to be qualitative factors, which means that they can take only the discrete values defined in the design, shown next (with the optimal settings highlighted).

 Name Level 1 Level 2 Level 3 Force 5 10 15 Power 40 50 60 Time 15 20 25 Temperature 155 160 165

If the factors can be treated as quantitative factors, meaning they can take any value within a range, further analysis using response surface methodology should be conducted to find the optimal settings for the manufacturing process.

* T. Hou, S. Chen, T. Lin and K. Huang, "An integrated system for setting the optimal parameters in IC chip-package wire bonding process," Int. J Adv Manuf Technology, 2006, 30, 247-253.

** G. Taguchi, S. Chowdhury and Y. Wu, Taguchi's Quality Handbook, Hoboken, New Jersey, Wiley, 2004.

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