Response Surface Methodology
Author: John M. Cimbala, Penn State University
Latest revision: 11 February 2008
Introduction
•
Taguchi design arrays were introduced previously as a way to hunt intelligently for an optimum point.
•
Taguchi’s design arrays are not infallible, however, and may not always be practical to set up, since they
work best with fairly
large changes
of the parameters.
•
There are
other
methods with which one can search or hunt for an optimum. One common method is called
response surface methodology
(
RSM
).
•
RSM is particularly useful for optimizing a
running
system
. For example, suppose a power plant is running
continuously. RSM can be used to optimize the efficiency or power output without large changes in the
operating parameters,
while continuing the operation without interruption
.
•
The main advantage of RSM is that it can be done “
on the fly
” with
small changes
of the parameters.
•
A good reference for most of this material is Design and Analysis of Experiments
, 4th ed., D. C.
Montgomery, John Wiley and Sons, QA279.M66 1996, Ch. 14.
Overview of the RSM Method
•
Suppose
y
=
y
(
a
,
b
,
c
, .
..), where
y
is the
outcome
or
result
or
response
that is to be optimized, and there are
n
parameters,
a
,
b
,
c
, .
.., which can be varied.
•
In these notes, it is assumed that the optimum
y
is the
maximum
y
. A similar analysis can be performed for
minimizing
y
.
•
The goal of RSM is to efficiently hunt for the optimum values of
a
,
b
,
c
, .
.. such that
y
is maximized
.
•
RSM works by the
method of steepest ascent
, in which the
parameters are varied
in the direction of
maximum increase of the response
until the response no longer increases.
•
RSM is best illustrated by examples. First, some simple
qualitative
examples are given with 1 and 2
parameters (
n
= 1 and
n
= 2), followed by a
quantitative
example with three parameters (
n
= 3).
•
Example for
n
= 1
: [
Note
: This is only a
qualitative
example.]
Given:
Response
y
=
y
(
a
). Here,
n
= 1 since there is only one parameter in the problem. The experiments
begin at some arbitrary operating point, i.e., at some value of
a
,
a
=
a
0
.
To do:
Apply RSM to find the optimum value of
y
, i.e., find the value of
parameter
a
where
y
is a maximum.
Solution:
a
y
a
0
o
Imagine a plot of
y
as a function of
a
, with initial operating condition
a
0
indicated by the
red dot
on the plot to the right.
o
Note that in an actual experiment, such a plot would not be available,
but is used here for illustrative purposes only.
o
With only one parameter (
n
= 1) as in this simple example, all one
needs to know is whether to search to the left or to the right. One of
these is the
direction of steepest ascent
.
o
To find this direction of steepest ascent, we measure
y
at some data
points
in the vicinity of
a
0
, as illustrated by the
blue dots
in the plot to
the right. Two other data points, in addition to that at
a
=
a
0
itself, are
sufficient for this simple case.