Minimum Aberration Fractional Factorial Designs With LargeN

Abstract

Xu has cataloged 165 minimum aberration (MA) regular fractional factorial (FF) designs with 2-levels and large run sizes N = 128 (m = 8-64 factors), N = 256 (m = 9-28, 109-119), N = 512 (m = 10-25, 238-246), N = 1024 (m = 11-24, 488-501), N = 2048 (m = 12-23), and N = 4096 (m = 13-24). Such an extensive catalog was produced because of an improved algorithm. We extend the catalog by 36 MA, 2level regular FF designs: N = 256 (m = 29-36, 100-108), N = 512 (m = 26-29), N = 1024 (m = 25-28), N = 2048 (m = 24-32), and N = 4096 (m = 25-26). Although such enumeration problems are notoriously difficult with increased N and/or m, we brought the newly solved problems within computational reach by changing the vital isomorphism check component of Xu's algorithm. Here we present a new, compact graph for solving regular design isomorphism problems and use the nauty program to solve the corresponding graph isomorphism problems. Supplemental appendices are available online. KEY WORDS: Bipartite undirected graph; Graph theory; Word-length pattern.

An Improved Algorithm for Constructing Large Fractional Factorial Designs

“…To illustrate the difference between the generation procedures, we compare the number of designs generated in creating a catalog of 128-run size, see (Table 2), both Xu and Schrivastava and Ding method's reduce the number of designs considered, for large factors the generation procedure of Xu introduce fewer designs in the intermediate set , as the table 2 show's for n > 9 our modified procedure gives best results in comparison with the other procedures; taking for example n = 11, the number of designs generated with Chen et al is of 711, for Xu 502 and for Schrivastava and Ding 703. For us the number of designs generated is from 219, the number is divided by 3.3, note that a comparison with results given by Xu method is also a comparison with [13] procedure because Ryan and Butlutuglo used the same generation method as in [15] . Ding method in creating a catalog of 4096-run with resolution 7, for n > 14 the number of designs considered for our method are reduced by 45%-81%.…”

“…[14] Extended some results from graph isomorphism literature to improve the design generation algorithm of [11]. Many other generation procedures were proposed in literature to produce FF-designs according to a particular criterion such as minimum aberration (MA) see for example: [10], [8] and [13].…”

RPackageFrF2for Creating and Analyzing Fractional Factorial 2-Level Designs

“…Several authors catalogued non-isomorphic regular fractional factorial 2-level designs, among them Chen, Sun, and Wu (1993), Mee (2005, 2006), Mee (2009), Xu (2009) and Ryan and Bulutoglu (2010). Regular fractional factorial 2-level designs can be parsimoniously catalogued by listing their generators via Yates matrix column numbers.…”

Experimental design

“…Depending on the criterion selected, the best OAs for a range of larger values of N and p have been determined without full enumeration. [23][24][25][26] Supersaturated designs have p ≥ N and hence necessarily risk bias even should a main effects model hold. They have nonetheless proven to be a valuable experimental tool when relatively few of the factors are expected to exert significant influence on the response y.…”