Publication date: 08/13/2020

## Coordinate-Exchange Algorithm

Custom Design constructs a design that seeks to optimize one of several optimality criteria. (See Optimality Criteria.) To optimize the criterion, Custom Design uses the coordinate-exchange algorithm (Meyer and Nachtsheim, 1995). The algorithm begins by randomly selecting values within the specified design region for each factor and each run to construct a starting design.

Suppose your study requires continuous factors, no factor constraints, and a main-effects model. An iteration consists of testing each value of the model matrix, as follows:

The current value of each factor is replaced by its two most extreme values.

The optimality criterion is computed for both of these replacements.

If one of the values increases the optimality criterion, this value replaces the old value.

The process continues until no replacement occurs for an entire iteration.

Appropriate adjustments are made to the algorithm to account for polynomial terms, nominal factors, and factor constraints.

The design obtained using this process is optimal in a large class of neighboring designs. But it is only locally optimal. To improve the likelihood of finding a globally optimal design, the coordinate-exchange algorithm is repeated a large number of times. Goos and Jones (2011, p. 36) recommend using at least 1,000 random starts for all but the most trivial design situations. The number of starting designs is controlled by the Number of Starts option. See Number of Starts. Custom Design provides the design that maximizes the optimality criterion among all the constructed designs.