# Optimal Exchange Methods

Optimal designs begin with a pseudo-random set of model points (runs) that are capable of fitting the designed for model. The initial selection can usually be improved by replacing a subset of the points with better selections. Stat-Ease uses one of five criteria to decide which replacements are better and up to two exchange methods to decide how they are replaced.

The Best Exchange uses Coordinate Exchange for half of the starting designs and Point Exchange for the other half. Both algorithms are given a chance to provide the best design.

## Coordinate Exchange

The Coordinate Exchange algorithm builds an approximately optimal design as follows:

1. Select a random initial set of p points, where p is the number of terms in the designed for model.

• Randomly pick each subsequent design point and evaluate if it increases the rank of the matrix. Continue this process until a full rank matrix is obtained.

2. Randomly select any extra model points.

3. Start the coordinate exchange algorithm.

• Calculate the current optimality criterion (OC)

• Sort the points by contribution to the OC.

• Starting with the worst point, move it along a set of directions in incremental steps.

• If the OC improves, change the point and move on to the next point in the list If the OC does not improve, retain the point and move on to the next point

• Once the entire list is exhausted, restart the algorithm

• If all points are retained then they form a locally optimal design.

• Repeat the algorithm several times to improve the odds of finding the globally optimal design.

4. Lack-of-fit points are added to the design to fill the largest gaps by selecting a group of points that maximizes the minimum distance to another point.

5. Replicates are chosen that best support the optimality criterion.

## Point Exchange

The Point Exchange algorithm builds an approximately optimal design as follows:

1. Define a candidate set of possible factor combinations.

2. Select a random initial set of p points from the above candidate set where p is the number of terms in the designed for model: