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JWBK119-19
Possible Applications 303
Table 19.3 Results of lean designs in incremental experimentation.
Estimated effect L 6 2 7−4 y 7 , y 8 absent L 7 2 7−4 y 8 absent 2 7−4 Regular data set
l 1 −11.40 −11.40 −10.875
l 2 0.65 −3.30 −2.775
l 3 −13.15 −17.10 −16.575
l 4 2.65 2.65 3.175
l 5 −23.35 −23.35 −22.825
l 6 0 −3.95 −3.425
l 7 0 0 0.525

Another possible lean design is one with seven experimental runs, or L 7 2 7−4 . In this
case only one factor needs to be assumed unimportant. If x 7 is such a factor and y 8 is
unavailable, then one can set
1
l 7 = (−y 1 + y 2 + y 3 − y 4 + y 5 − y 6 − y 7 + ˆy 8 ) = 0. (19.9)
4
Hence
ˆ y 8 = y 1 − y 2 − y 3 + y 4 − y 5 + y 6 + y 7 = 36.6, (19.10)
with which all the other l i values can be computed.
The results of analysis of the above two lean designs are exhibited in Table 19.3,
along with the results obtained from the full 2 7−4 experimental data from the filtration
study. It is evident from the table that, using an L 6 2 7−4 design, the investigator would
arrive at exactly the same four tentative conclusions associated with the data from
6
the regular 2 7−4 design (p. 426 of Box et al. ). With one more experimental run -- that
is, with an L 7 2 7−4 design -- there would still be no change in the interpretation. This
shows, in a striking manner, that with a considerably smaller amount of experimental
effort, lean design works extremely well when the investigator is able to make a good
prior judgment concerning the relative importance -- or rather, unimportance -- of the
factors under study.
Even if the judgment of the investigator is somewhat off the mark, especially in
the presence of unsuspected interactions, the results of a lean design can still be
gradually improved upon by the addition of one further experimental run at a time.
Thus the filtration plant study can start with a lean design of L 6 2 7−4 to obtain some
provisional findings, followed by the generation of response y 7 to produce the results
corresponding to an L 7 2 7−4 design, and finally y 8 to arrive at what a regular 2 7−4
design can reveal about the plant behavior. Such an incremental experimentation or one-
additional-run-at-a-time scheme, has the same merit as that advocated for sequential
fold-over designs, namely earlier availability of first results, and improvement of such
results as new data becomes available.

19.6 POSSIBLE APPLICATIONS
Lean design is useful in circumstances such as those described below.

1. It is imperative to have only the minimum number of experimental runs in view
of, for example, expensive prototypes, high costs of destructive tests, or costly
disruptions to regular production.