Minimum cost trend free 16-run foldover designs
Main Article Content
Abstract
This paper proposes new 16-run foldover designs aimed at minimizing the number of level changes. The method involves selecting n - p independent columns from the main effects and interaction effects of a 24 full factorial experiment while avoiding run duplication to construct a 2n-p fractional factorial design. The remaining p columns are then generated using these selected independent columns to maintain cost efficiency. The number of level changes and trend-free factors are computed for all possible fold-over plans for both standard and new designs. The performance of the new designs is compared to standard designs proposed by Li and Lin (2003), Cheng and Steinberg (1991) and Coster (1993) using the criteria of minimum level change, maximum trend-free factors and uniformity exhibiting improved performance and cost-effectiveness.
Article Details
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
References
1. Adekeye, K. S., & Kunert, J. On the comparison of run orders of unreplicated 2k-p designs in the presence of a time trend. Metrika, 63, 257-269 (2006). https://hdl.handle.net/10419/22594
2. Bhowmik, A., Varghese, E., Jaggi, S., & Varghese, C. On the generation of factorial designs with minimum level changes. Communications in Statistics-Simulation and Computation, 51(6), 3400-3409 (2020). https://doi.org/10.1080/03610918.2020.1720244
3. Cheng, C. S. Run orders of factorial designs. Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Keifer, H, 619-633 (1985). https://scholar.google.com/scholarhl=en&as_sdt=0%2C5&q=3.+Cheng%2C+C.+S.+%281985%29.+Run+orders+of+factorial+designs.+Proceedings+of+the+Berkeley+Conference+in+Honor+of+Jerzy+Neyman+and+Jack+Keifer%2C+H%2C+619-633.&btnG=
4. Cheng, C. S., &Jacroux, M. The construction of trend-free run orders of two-level factorial designs. Journal of the American Statistical Association, 83(404), 1152-1158 (1988). https://doi.org/10.1080/01621459.1988.10478713
5. Cheng, C. S., Martin, R. J., & Tang, B. Two-level factorial designs with extreme numbers of level changes. Annals of statistics, 1522-1539 (1998). https://doi.org/10.1214/aos/1024691252
6. Cheng, C. S., & Steinberg, D. M. Trend robust two-level factorial designs. Biometrika, 78(2), 325-336 (1991). https://doi.org/10.1093/biomet/78.2.325
7. Coster, D. C. Tables of minimum cost, linear trend-free run sequences for two-and three-level fractional factorial designs. Computational statistics & data analysis, 16(3), 325-336 (1993). https://doi.org/10.1016/0167-9473(93)90133-E
8. Coster, D. C., & Cheng, C. S. Minimum cost trend-free run orders of fractional factorial designs. The Annals of Statistics, 16(3), 1188-1205 (1988). https://doi.org/10.2307/2241626
9. Cox, D. R. Some systematic experimental designs. Biometrika, 38(3/4), 312-323 (1951). https://doi.org/10.2307/2332577
10. Daniel, C., & Wilcoxon, F. Factorial 2p-qplans robust against linear and quadratic trends. Technometrics, 8(2), 259-278 (1966). https://doi.org/10.1080/00401706.1966.10490346
11. Dickinson, A. W. Some run orders requiring a minimum number of factor level changes for the 2⁴ and 2⁵ main effect plans. Technometrics, 16(1), 31–37 (1974). https://doi.org/10.2307/1267489
12. Draper, N. R., & Stoneman, D. M. Factor changes and linear trends in eight-run two-level factorial designs. Technometrics, 10(2), 301-311 (1968). https://doi.org/10.2307/1267045
13. Fang, K. T., Lin, D. K., Winker, P., & Zhang, Y. Uniform design: theory and application. Technometrics, 42(3), 237-248 (2000). https://doi.org/10.1080/00401706.2000.10486045
14. Fang, K. T., Lu, X., & Winker, P. Lower bounds for centered and wrap-around L2-discrepancies and construction of uniform designs by threshold accepting. Journal of Complexity, 19(5), 692-711 (2003). https://doi.org/10.1016/S0885-064X(03)00067-0
15. Fang, K. T., Ma, C. X., & Mukerjee, R. Uniformity in fractional factorials. In Monte Carlo and Quasi-Monte Carlo Methods 2000: Proceedings of a Conference held at Hong Kong Baptist University, Hong Kong SAR, China, November 27–December 1, 2000. 232-241. Berlin, Heidelberg: Springer Berlin Heidelberg (2002). https://doi.org/10.1007/978-3-642-56046-0
16. Hickernell, F. A generalized discrepancy and quadrature error bound. Mathematics of computation, 67(221), 299-322 (1998). https://www.jstor.org/stable/2584985
17. Joiner, B. L., & Campbell, C. Designing experiments when run order is important. Technometrics, 18(3), 249-259 (1976).https://doi.org/10.2307/1268734
18. Li, W., & Lin, D. K. J. Optimal foldover plans for two-level fractional factorial designs. Technometrics, 45(2), 142–149 (2003). https://doi.org/10.1198/004017003188618779
19. Qin, H., & Fang, K. Discrete discrepancy in factorial designs. Metrika, 60(1) (2004). https://doi.org/0.1007/s001840300296
20. Singh, P., Thapliyal, P. & Budhraja, V. Construction of Trend free Run Orders for Orthogonal Arrays using linear codes. International Journal of Engineering and Innovation Technology.3(1), 69-73 (2014).
21. Singh, P., Thapliyal, P., & Budhraja, V. Construction of linear trend free fractional factorial designs using linear codes. Int. J. Agricult. Stat. Sci, 12(1), 13-19 (2016a).https://connectjournals.com/pages/articledetails/toc024946
22. Singh, P., Thapliyal, P., & Budhraja, V. A technique to construct linear trend free fractional factorial design using some linear codes. International Journal of Statistics and Mathematics, 3(1), 73-81 (2016b). https://www.researchgate.net/publication/307588875_Construction_of_linear_trend_free_fractional_factorial_designs_using_linear_codes
23. Singh, P.,Thapliyal, P., &Budhraja, V. Construction of trend free run orders for orthogonal arrays using linear codes. International Journal of Engineering & innovation technology, 3(1), 69-73 (2013). https://www.ijeit.com/Vol%203/Issue%201/IJEIT1412201307_13.pdf
24. Thapliyal, P., & Budhraja, V. On the Construction of Trend Free Run Order of the Two-Level Factorial Design Using BIBD. American Journal of Theoretical and Applied Statistics, 9(6), 263 (2020). https://doi.org/10.11648/j.ajtas.20200906.11
25. Varghese, E., Bhowmik, A., Jaggi, S., Varghese, C., & Lall, S. On the construction of response surface designs with minimum level changes. Utilitas Mathematica, 110, 293-303 (2019).https://www.researchgate.net/publication/336737157_On_the_Construction_of_Response_Surface_Designs_with_Minimum_Level_Changes