# Modules of the Gendex DOE Toolkit 2015

The latest version of the Gendex DOE Toolkit has 14 modules. The function of each module is described below. You can get access to the quick start guide of each module by clicking on the corresponding module in the left panel.

#### ALPHA

Patterson & Williams (1976a) introduced a new class of resolvable incomplete block design called α-designs. Since their introduction, α-designs have become popular among designers of experiments due to two main reasons: (i) α-designs are available for many (r,k,s) combinations where r is the number of replicates, k is the block size and s is the number of blocks per replicate (the number of treatments v=ks); (ii) the computer revolution has brought the PC to the desk of most designers of experimenters. As such, the flexibility of the design has succeeded computational simplicity as their criterion in design selection. ALPHA is a Gendex program for constructing optimal or near-optimal α-design. ALPHA can handle up to 10,000 treatments. Designs constructed by ALPHA can be used as column components of resolvable row-column designs (see Nguyen & Williams 1993). ALPHA uses a 2-stage optimization process. Each stage of the optimization process uses an algorithm similar to the cyclic-coordinate exchange algorithm described in Nguyen (2002) . The algorithm uses extensively some theoretical results described in Patterson & Williams (1976b).

#### IBD

An incomplete block design (IBD) of size (v,k,r) is an arrangement of v treatments set out in blocks of size k (<v) such that each treatment is replicated r times. We confine to binary IBDs, i.e. IBDs in which no treatment occurs more than once in a block. An IBD is said to be a balanced IBD, or BIBD, if every pair of treatments occurs in exactly blocks. An IBD is said to be t-resolvable if its blocks can be divided into subsets and each subset is an IBD of size (v,k,t). IBD is a Gendex program for constructing optimal or near-optimal IBDs (resolvable and non-resolvable). The optimality criterion and the algorithm used in IBD is discussed in Nguyen (1993, 1994). Designs produced by IBD are comparable to α-designs of Patterson & Williams (1976) and generalized cyclic designs of Hall & Jarrett (1981) in terms of the efficiency factor E of the design. These designs, in turn, can be used as column components of row-column designs (Nguyen & Williams 1993; Nguyen 1997).

#### CIBD

Cyclic incomplete block designs (IBDs) are IBDs generated by the cyclic development of one or more suitably chosen initial blocks. Cyclic IBDs accounts for a large number of balanced IBDs or BIBDs in Fisher & Yates (1963) and Rao (1961). They also provide efficient alternatives to many partially balanced IBDs, or PBIBDs, catalogued in Clatworthy (1973). When the number of replications r is equal to or is a multiple of the block size k, cyclic IBDs render automatic elimination of heterogeneity in two directions. CIBD is a Gendex program for generating optimal or near-optimal cyclic IBDs. CIBD can handle up to 10,000 treatments. Designs constructed by CIBD can be used as column components of row-column designs (see Nguyen 1997). CIBD uses a 2-stage optimization process. Each stage of the optimization process uses an algorithm similar to the cyclic-coordinate exchange algorithm described in Nguyen (2002).

#### RCD

A (non-resolvable) row-column design (RCD) is an arrangement of v treatments set out in kxb array such that each treatment is replicated r times (vr=kb). We say this RCD is of size (v,k,b). RCD is a Gendex program for constructing optimal or near-optimal RCDs. The approach adopted by RCD is to permute the treatments within the blocks of an optimal or near-optimal block design used as the column component of the RCD. The optimality criterion and the algorithm used in RCD are discussed in Nguyen (1997).

#### RRCD

A resolvable row-column design (RCD) is an arrangement of r kxs arrays each of which is a complete replicate of v=ks treatments. We say this resolvable RCD is of size (r,k,s). RRCD is a Gendex program for constructing optimal or near-optimal resolvable RCDs. The approach used by RRCD is to permute the treatments within the blocks of a resolvable incomplete block design (IBD) used as the column component of the RCD. This IBD can be constructed by either the IBD module or the ALPHA module of the Gendex toolkit. The optimality criterion and the algorithm implemented in RRCD are discussed in Nguyen & Williams (1993).

FEADO (Fedorov exchange algorithm for D-optimal experimental designs) constructs D- and G-optimal 2-level, 3-level and mixed-level fractional factorial designs and response surface designs. FEADO can also construct designs for constrained regions including mixture designs by choosing a subset of runs from a given set of candidate runs. There is no limit on the number of runs in the candidate set. FEADO uses a fast Fedorov's exchange algorithm described in Nguyen & Miller (1992), Miller & Nguyen (1994) and Nguyen & Piepel (2004).

#### MIGA

MIGA is a program for constructing minimum G-aberration designs. The minimum G-aberration criterion proposed by Tang & Deng (1999) is a generalized version of the popular minimum aberration criterion of Fries & Hunter (1980). Designs constructed by MIGA include both regular and non-regular 2-level fractional factorial designs.

#### SOD

SOD is a program for constructing constructs 3-level second-order response surface designs for m factors in n runs Dmxn where each factor takes only levels -1, 0 and +1. The constructed designs include BBD, CCDs, small CCDs, etc. The detailed account of the SOD algorithm appears in Nguyen & Lin (2011).

#### CUT

CUT is a Gendex program for multi-dimensional blocking fractional factorial designs and response surface designs. The CUT approach to blocking a design is to find a suitable unblocked design (constructed by other Gendex modules such as MIGA and SOD) and allocate the n runs of this design to blocks, or rows and columns, etc. The algorithm which implements the CUT approach is the extension of the one appeared in Nguyen (2001).

#### RAT

RAT is a Gendex program for constructing trend-free fractional factorial designs and response surface designs, i.e. designs which are robust against time trends. RAT uses an algorithm which is described in Nguyen (2014).

#### NOA

A strength 2 orthogonal array (OA) of size n with k si-level columns (i = 1,...,k), denoted by Ln(s1...sk) is an n x k matrix in which all possible combinations of levels in any two columns appear the same number of times (Rao 1947). In a near-OA L'n(s1...sk), to reduce the run size, the orthogonality of some pairs of columns is necessarily sacrificed. NOA is a Gendex program for constructing mixed level OA, near-OAs and supersaturated designs. Details of the NOA algorithm which uses this optimality criterion is discussed Nguyen & Liu (2007).

#### LHD

Latin hypercubes (LHs) were designs introduced by McKay, Beckman & Conover (1979) for computer experiments. An nxk LH can be represented by a design matrix Xnxk with n rows (runs) and k columns (factors), each of which includes n uniformly spaced levels. An LH is called an orthogonal LH (OLH) if each pair of columns of this LH has zero correlation. LHD is a Gendex program for constructing near-OLHs with good space-filling property using the algorithm described in Nguyen & Lin (2012). The near-orthogonal OLHs constructed by LHD are quite good compared to those of Cioppa & Lucas (2007) and those in http://www.ams.sunysb.edu/~kye/olh.html with respect to both orthogonality and space-filling properties.

#### SPLIT

Several industrial experiments are set up in second-order split-plot designs (SPDs). SPLIT is a program for constructing D-efficient equivalent-estimation second-order SPDs (EE-SPDs) by interchanging the levels of the SP factors within each WP. The produced EE-SPDs are not only highly D-efficient but also attractive from the experimenters' point of view as their information matrices are close to the structure possessed by the ones of popular designs such as BBDs, CCDs and BBD- and CCD-based SPDs (Parker et al, 2006, 2007). In addition, they could improve all EE-SPDs with non-integer levels reported in Jones & Goos (2012). The detailed account of the SPLIT algorithm appears in Nguyen & Pham (2013).

#### SUDOKU

Sudoku is a logic-based number-placement puzzle. The objective is to fill a 9x9 grid so that each column, each row, and each of the nine 3x3 boxes (also called blocks or regions) contains the digits from 1 to 9 only one time each. The puzzle provides a partially completed grid. SUDOKU is a Sudoku puzzle solver and generator.