AI summary

This paper discusses various experimental procedures for estimating the optimal point on a response surface using techniques such as multi-factor experiments, factorial designs, and sequential one-factor-at-a-time procedures, and presents the use of the path of steepest ascent to get to a “near-stationary” region.

Abstract

This paper discusses various experimental procedures used to estimate the optimal point on a response surface and to explore the nature of the response surface in the vicinity of this optimum. Multi-factor experiments were first set up to investigate one factor at a time; then Fisher and Yates introduced the complete factorials for field experiments, plus confounded arrangements for incomplete blocks designs. More recently, fractional replication designs have been introduced to cut down the size of the experiments. Hotelling devised methods of locating the optimal point using a single factor. Friedman and Savage outlined a sequential one-factor-at-a-time procedure when several factors are involved. Box and Wilson present a method of locating the optimum and of exploring the response surface in which many factors are varied at the same time. They present the use of the path of steepest ascent to get to a "near-stationary" region if the experimenter starts at a point far removed from it. When the experimenter is near such a region, they present the use of a composite design to estimate quadratic and interaction effects. The nature of the response surface is explored by the use of a canonical transformation. The usefulness of these sequential procedures in various experimental situations is discussed.