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. Design-Expert 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.

  • Start with a random coordinate (point) within the design space.

  • 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.

  1. Randomly select any extra model points.

  2. 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.

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

  2. Additional centroids, if any, are added.

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:

  • Start with a random point from the candidate set.

  • Randomly pick each subsequent design point. If a new point increases the

  • rank of the matrix point it is added to the starting bootstrap. Continue

  • this process until a full rank matrix is obtained.

  1. Randomly select any extra model points.

  2. Perform exchange steps:

    • A 1-point exchange step consists of adding to the current design the point

    • in the candidate list that improves the optimality criterion the most and

    • then deletes from the augmented design the point that improves the

    • selection criterion the least. A 2-point exchange step adds two points in

    • sequence, and then deletes two points. An n-point exchange step adds and

    • deletes n points.

    • Perform 1-point exchange steps until there is no improvement in the

    • design. Then perform 2-point exchange steps, and so on until up to

    • 10-point exchange steps show no improvement to the optimality criterion.

    • If at any point, there is improvement, start over with 1-point exchanges.

    • “P”, the number of excursions, can be set by clicking the options button.

Point exchange is used to restrict the available runs to a specified candidate set.