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Potential of Lean Design 299
statistics, or values of factor effects, that can be computed is equal to the number of
experimental runs. Technically, this is to say that the degrees of freedom associated
with the computed effects equal the degrees of freedom carried by the raw data used
3
in the analysis. Thus in an unreplicated 2 full factorial design, eight response values
y 1 , y 2 ,..., y 8 are used to generate eight statistics E 0 , E 1 , E 2 , E 3 , E 12 , E 13 , E 23 , and
E 123 . With the eight response values from a fractional factorial design, say of the type
2 4−1 , eight statistics are generated to represent a collection of main and interaction
effects -- 16 in total -- inherent in the subject under study. If all interaction effects
are of interest, there will be confounding of two effects in every statistic; however, if
interaction effects are non-existent, four clear main effects E 1 , E 2 , E 3 , E 4 , in addition
to the mean E 0 , can be obtained.
Without the interactions, since E 0 , E 1 , E 2 , E 3 , E 4 account for five degrees of free-
dom, the three remaining degrees of freedom are arguably not really needed. Taguchi
analysts would create an error sum of squares to absorb the three degrees of freedom,
but this is an expediency that does not result in something as reliable and useful as a
pure error sum of squares obtainable from replicated experimental runs. Since these
three redundant degrees of freedom must originate from three response values, one
can conclude in turn that three out of eight such response values are, in this sense,
unnecessary. Hence only five of the experimental runs, not eight, need actually be
carried out. An experiment formulated along this line of thinking -- based on an or-
thogonal design not meant to be carried out in full, thus achieving economy in the
required experimental effort -- is one of lean design.
It is clear from the above discussion that lean design can be said to be ‘more Taguchi
than Taguchi’. First, one is supposed to know what to ask for in an experimental data
analysis -- an a priori listing of main and interaction effects to be obtained from the
experiment. Secondly, since all other interactions are assumed to be non-existent,
confounding will not be a concern. Furthermore, as long as the total number of effects
asked for is less than what the size of the design matrix would provide, one has the
option -- not offered in ‘regular’ Taguchi methods -- not to obtain the complete set of
response values: for example, if only four effects are needed, one could opt to collect
as few as five observations out of an eight-run experiment. (Of course six, seven, or
eight can also be collected if desired.)
For ease of reference, a lean design can be represented by the notation L n 2 k−p , where
k
2 k−p has the usual meaning and can be replaced by 2 when a full factorial is used,
and n represents the number of experimental runs intended to be actually carried out.
Note that n ≤ 2 k−p . Since E 0 always has to be provided for, n is at least equal to one
more than the total number of main effects and interaction effects requested by the
investigator.

19.4 POTENTIAL OF LEAN DESIGN

For an appreciation of the potential of lean design, the case of the eight-run experiment
is fully explored in Table 19.1. For example, with four factors, a requirement for four
main effects but with total disregard for interaction effects can be met with a design
of type L 5 2 4−1 . For a regular eight-run 2 5−2 design for five factors, if no interactions
are needed, an L 6 2 5−2 design can be used instead. Similarly, for a 2 6−3 design for six
factors, a design of type L 7 2 6−3 is available. Considerable savings in experimental