Three-Level Simplex Designs and Their Use in Sequential Experimentation - ASQ

Three-Level Simplex Designs and Their Use in Sequential Experimentation

In this paper three-level simplex designs of k+1 runs for k factors are presented. Each simplex design is composed of k treatment combinations from a two-level factorial design, plus an additional base run that represents a third level for each factor. These orthogonal, first-order designs are simple to construct. Furthermore, simplex designs can be augmented to construct second-order simplex sum designs. Such designs are particularly attractive when the experimental region of interest is spherical rather than a hypercube. It is also noted that the k treatment combinations in each simplex design coincide with the axial points in the second-order, asymmetric composite designs proposed by Box and Wilson (1951) and later discussed by Lucas (1974). Including the base point permits the inclusion of a block effect in the fitted second-order model.

Key Words: Asymmetric Composite Design, Augmented Pair Design, Response Surface Methodology, Second-Order Design, Simplex Sum Design.

By ROBERT W. MEE, University of Tennessee, Knoxville, TN 37996-0532

INTRODUCTION

SIMPLEX designs are saturated, orthogonal first-order designs based on the vertices of regular geometric gures (e. g., see Myers and Montgomery (1995, section 7.3.3)). Crosier (1996) proposed sym-metric simplex designs, that is, simplex designs oriented so that each factor takes the same set of levels. While Crosier s designs were for practitioners an improvement over other common orientations, his de-signs require five or six factor levels for most values of the number of factors, k. Such designs are rather complicated for fitting the simple first-order model.

A symmetric simplex design with three levels can be constructed for any value of k by combining k axial points and a single base point. Let the axial point for the ith factor be with and , where is a scale factor and is a location parameter to be specified later. The base point is , where

. (1)

The positive and negative solutions to Equation (1) are denoted by + and -, respectively. Either solution to Equation (1) produces a saturated, orthogonal main effects design. For instance, for k = 3 and µ = 0, the proposed simplex design corresponding to the solution - = -1/3 is

whereas the solution corresponding to + = 1 is

Note that, for k = 3, the + design has only two levels ( since + = 1) and is equivalent to the usual fractional factorial design. This is the only such case where the proposed simplex does not have three levels, since, as k increases, both + and - monotonically approach zero ( refer to Table 1).

When using this simplex design as an initial experiment, for convenience one may specify the location and scale parameters as = 2and µ = 1 so that the k axial points take on the levels ± 1. For the solution + with k > 2, the third (base) level is within [1 , 1]. The extreme levels of the actual experimental factors are thus assigned to the coded levels ± 1. For the solution, the base location represents the minimum level (< -1). Further details regarding the use of a three-level simplex as the initial design for


sequential experimentation are presented in the next section.

It would be expected that such a simple simplex orientation would have been noted previously. Spendley, Hext, and Himsworth (1962) did present a three-level simplex corresponding to the - solution above. Spendley et al. (1962) set the base location to equal (0..., 0) and defined the other treatment combinations of the simplex to be a distance 1 from the base, with two positive levels for each factor. Box (1952) discussed the consequences of different rotations of a simplex design, but does not mention a three-level orientation.

As shown by Box and Behnken (1960), any simplex design can be augmented to construct a second-order rotatable design by adding all sums of the simplex runs, appropriately scaled. For example, for the k = 3 simplex with + = 1, the six sums of pairs of runs are (where the radius multiplier = 8 1/4 for rotatability), and the four sums of triplets form the foldover of the original 2 3-1 design. Summing all four of the original simplex runs produces the center point, which we replicate n times. This (4 + 6 + 4+ n) run design corresponds to the well-known central composite design. If one takes another simplex for k = 3 having more than two-levels, a different rotatable second-order design is obtained that does not correspond to the central composite design. In a subsequent section, Box and Behnken s rotatable simplex sum designs are summarized, and a class of (non-rotatable) simplex sum designs with the minimal number of treatment combinations is introduced. These new saturated second-order designs are akin to Morris' (2000) augmented pair designs, except that they have more than three levels.

The simplex and simplex sum designs are best suited for spherical regions. When the region of interest is a hypercube, the attractiveness of simplex designs is diminished. The contrast between simplex designs and saturated two-level designs over the hypercube is addressed in a separate section.

Finally, the three-level simplex design can also be used to augment an initial two-level factorial or fractional factorial design. When the axial points are located in the direction of one of the original factorial points, the factorial and simplex designs together form an asymmetric composite design (ACD) (Lucas (1974)). It was this application which led to the discovery of the simplex orientation presented here. After noting that the k axial points for an ACD are equidistant from one another, the base point was positioned to achieve a simplex design. The later section on ACDs contains more details (see also Mee (2001)).

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