A HEURISTIC METHOD FOR PLANNING TWO-LEVEL FRACTIONAL FACTORIAL EXPERIMENTS USING BASIC QUATERNARY DESIGN TABLE

啟發式四分位設計表於兩水準部分實驗設計之應用

10.6220/joq.2017.24(4).05

戴貞德(Jen-der Day)

fractional factorial design ； graphical aid ； quaternary design table ； heuristic method ； 部份實驗設計 ； 輔助圖 ； 四進位設計表 ； 啟發式法

品質學報

24卷4期（2017 / 08 / 30）

300 - 310

英文

Graphical aids are most useful for planning two-level fractional factorial experiments that allow un-confounded estimation of all main effects and some specified interactions in a requirements set under the assumption that all un-specified interactions are negligible. Some graphical aids such as Taguchi's linear graphs, interaction graphs, and a graph-aided method have been published, and they are easy to use and visually appealing. However, any graph-aided method becomes impractical due to a large number of graphs and visual complexity when the run size is large. In our earlier work, the geometrical design was chosen as a standard design matrix for a two-level fractional factorial design. Then, based on the quaternary number representation system for column numbers of a geometrical design, the basic quaternary design table (BQDT) was proposed for planning two-level fractional factorial experiments. In this article, by applying the BQDT, a heuristic method is proposed to solve the above estimation problem for planning two-level fractional factorial experiments. The arc welding example with 9 two-level factors and 16 runs is illustrated to compare three methods of linear graph, interaction graph, and our heuristic BQDT method, and the result shows that our method has minimum aberration. Two defect examples of Greenfield's algorithm are also improved by our method. The most important advantages of the proposed heuristic BQDT method are ease to use, visually appealing, and possible for large scale problem extension.

當未指定的交互作用是可忽略時，輔助圖是估計兩水準部份實驗中未交絡的主效果與二階交互作用的有效工具。文獻中的田口點線圖、交互作用圖與輔助圖法均有視覺與易用的特性，然而當實驗規模較大時這些輔助圖則有實用上的缺陷。先前我們依據幾何設計而發展出的四分位設計表以利規劃兩水準實驗設計，本文將進一步發展出啟發式四分位設計表來解決上述估計未交絡的主效果與二階交互作用的問題。電弧焊範例中含有9個兩水準因子與16個實驗次數，運用此例比較三種方法田口點線圖、交互作用圖與本文的啟發式法，結果發現啟發式得到變異最小化的最佳設計。另外，Greenfield演算法中的兩個缺失也可透過啟發式法輕易解決。綜上，啟發式四分位表具有視覺性與易用性，且適用於大規模的兩水準實驗設計。

1. Plackett, R. L. and Burman, J. P., 1946, The design of optimum multifactorial experiments, Biometrika, 33(4), 305-325.
2. Box, G. E. P.,Wilson, K. B.(1951).On the experimental attainment of optimum conditions.Journal of the Royal Statistical Society. Series B (Statistical Methodological),13(1),1-45.
3. Chen, J.,Sun, D. X.,Wu, C. F. J.(1993).A catalogue of two-level and three-level fractional factorial designs with small runs.International Statistical Review,61(1),131-145.
4. Day, J.-D.,Tsai, H.-T.(2013).Comparison of useful characteristics among various two-level design matrices.Proceeding of the 16th QMOD Conference on Quality and Service Sciences ICQSS 2013,Portorož, Slovenia:
5. Day, J.-D.,Tsai, H.-T.(2013).Basic quaternary design table using geometrical design.Proceeding of 2013 International Conference on Technology Innovation and Industrial Management,Phuket, Thailand:
6. Day, J.-D.,Tsai, H.-T.(2013).Joint quaternary design tables for geometrical designs.Proceedings of the 19th ISSAT International Conference on Reliability and Quality in Design,Honolulu, HI.:
7. Franklin, M. F.,Bailey, R. A.(1977).Selection of defining contrasts and confounded effects in two-level experiments.Applied Statistics,26(3),321-326.
8. Fries, A.,Hunter, W. G.(1980).Minimum aberration 2k-p designs.Technometrics,22(4),601-608.
9. Greenfield, A. A.(1976).Selection of defining contrasts in two-level experiments.Journal of the Royal Statistical Society. Series C (Applied Statistics),25(1),64-67.
10. Greenfield, A. A.(1978).Selection of defining contrasts in two-level experiments - a modification.Journal of the Royal Statistical Society. Series C (Applied Statistics),27(1),78.
11. Kacker, R. N.,Tsui, K.-L.(1990).Interaction graphs: Graphical aids for planning experiments.Journal of Quality Technology,22(1),1-14.
12. Li, C. C.,Chiou, K. M.(1989).Some explanation and evaluation of the linear graphs of Taguchi's orthogonal arrays L8 (27) and L16 (215).Journal of the Chinese Institute of Industrial Engineers,6(1),43-47.
13. Li, C. C.,Washio, Y.,Iida, T.,Tanimoto, S.(1990).New linear graphs for orthogonal array L16 (215).Journal of the Chinese Institute of Industrial Engineers,7(1),17-23.
14. Taguchi, G.(1987).Systems of Experimental Design.White Plains, NY:UNIPUB/Kraus.
15. Taguchi, G.,Wu, Y.(1980).Introduction to Off-line Quality Control.Tokyo, Japan:Central Japan Quality Control Association.
16. Tsai, H. T.(1999).Number representation algorithm for obtaining two-factor interaction columns in two-level orthogonal arrays.International Journal of Quality and Reliability Management,16(6),552-561.
17. Wu, C. F. J.,Chen, Y.(1992).A graph-aided method for planning two-level experiments when certain interactions are important.Technometrics,34(2),162-175.

1. Yu-Lin Han,Tsai-Hsin Cheng,Jen-der Day,Hsin-Lu Liu,Hsien-Tang Tsai(2023)。Binary Column Assignment Method for Two-Level Minimum Aberration Fractional Factorial Designs。品質學報，30(2)，67-88。
2. Yu-Lin Han,Tsai-Hsin Cheng,Jen-der Day,Hsin-Lu Liu,Hsien-Tang Tsai(2023)。Closed-Form Formulae of Wordlength Pattern for Saturated Resolution IV and III Designs and Their Relationships。品質學報，30(1)，13-25。