Let us now discuss the meaning and the potential validity of the hypotheses of the previous Sections. Such a discussion is obviously the key point when trying to induce a practitioner to adopt any academic formula to conduct his own business.
The first point to be mentioned concerns what is modeled. When using
a probability distribution 
, we can model our
lack of actual knowledge concerning the future demand. We may, for
example, think that a better knowledge (at ordering time) can be reached
but that its cost would be greater than the additional benefits resulting
from this additional knowledge. Another point of view is that markets
are intrinsically wild so that the probability function rather models
the very nature of market [Corker 1986]. This point, of philosophical
nature, is obviously an open question, but nevertheless cannot be
omitted.
The second point concerns what experimental procedure can be used
to determine . A "Gedachte Experiment"
is as follows: starting with a great number of exact copies of the
actual world, put different orders in these worlds, inducing them
to evolve (independently) in different manners and observe what happens
at selling time. One cannot escape this point of view by considering
approximations obtained from times series, since only ergodicity can
justify such approximations (without mentioning the fact that actual
times series are quite ever too short to conclude, even assuming ergodicity).
A third point is that the actual demand cannot be observed (i.e. measured,
even afterwards) when the demand overflows the inventory. In such
a case, the only actual knowledge is that an overflow has occurred.
Obtaining any quantity concerning the demand (i.e. the whole range
of the distribution) only from what
happens in the range 
(i.e. from only
the observed demand) cannot be done in a 'distribution free' manner.
Even the mean 
cannot be guessed that way.
This fact is a key point when discussing the meaning of the Scarf's
theorem. In the original paper [Scarf 1958], the author only states:
"Let 
be fixed. Then...". Afterwards,
other authors have presented this theorem as a 'distribution free'
result, on the basis that the "Scarf's rule" is
issued from a family of distributions rather than from a given distribution.
But, in our opinion, this presentation is too optimistic.
Moreover, any 'educated guess' of the 
part of the reality
cannot be 'cost free' either. And this cost must be incorporated into
the total income we want to optimize. For example, if the newsboy
increases his order quantity to observe more demand, he will certainly
increases his knowledge of the demand, but the core question was increasing
his income.
Let us now take an example and examine what can be done when historical
data are available. For example, if the last 
orders were 
and the actual past sales were:
But, as discussed before, nothing can be said on the really interesting
parameters, namely 
.
To estimate these parameters, we must recreate the missing data, i.e.
the exact values reached by the demand when it has overflowed the
inventory. This can be done in many ways but, in any case, the figures
obtained are only fictional, and have an extra cost. Let us assume
the following non fully observed demands:
In tab: best-decisions, these estimators have been used
to compute the best decisions against three distributions and four
families. The values 
and either 
(left part)
or 
(right part, not otherwise commented) have been
used. The first five lines describe what are the consequences of "assuming
an exact knowledge of ", while the last two
are relative to the consequences of "assuming an exact knowledge
of 
".
If nothing else than 
is assumed, the
Scarf's robust decision ensures a expected gain of 
. The additional
value of an information concerning the shape of is given in
tab:value-of-shape-sigma. It should be noticed that these
values are not obtained by subtracting the corresponding gains from
2 (when you are not aware that has
a given shape then, nevertheless, you are not playing the "Scarf's
rule" against the worst "two Dirac's" distribution,
but against this given ). It can be seen that, in our example,
the value of these informations relative to the shape is negligible.
In tab:value-of-sigma-delta-Dirac, the value of an information relative to which is the right measure of the dispersion is computed (assuming that the play is against the "two Dirac's" distributions). It should be noticed that these values are significantly greater than those of tab:value-of-shape-sigma.
|
In fact, an exact knowledge of and of either 
or 
can hardly be assumed. If we reexamine the former data and assume
that the successive demands were independent and identically distributed
then, without any other assumption, the distribution of 
is roughly normal, with mean and variance 
,
while the distribution of 
is also roughly normal,
with mean 
and variance 
,
where 
is the fourth centered moment. In other words, our knowledge
is :
are the coverage factors.
From the Student's distribution, we know that 
is an optimistic
choice, and all what we actually know is that should stay somewhere
in an interval at least as wide as 
, while
should stay somewhere in an interval at least as wide as
.
In tab:value-of-mu-knowing-sigma, the information value
of the coverage factor 
is discussed when assuming that
is exactly known. The values obtained in this Table and
in tab:comparison-delta-sigma are casually of the same
magnitude, but they are not of the same nature since those of tab:value-of-mu-knowing-sigma
do depend from the size of the history while the others don't.
It must be noticed that having historical data from the last
independent periods is out of question in many domains, the textile
industry among them. Moreover, disposing of a 
history will
only reduce the uncertainties by a factor 
and the influence of
these uncertainties will remain dominant. In such a case, the search
of a robust solution must be enlarged to the family of all the distributions
whose parameters fall in the intervals of uncertainty.
As said before, the information values concerning the shape of the
distribution computed in sub:Numerical-example were quite
negligible due to the "not so great" value of 
.
Let us now consider what will happen if the dispersion is multiplied
by a proportionality factor 
. All the deviations 
(except from those relative to lognormal shapes) will be multiplied
by , and therefore the differences between all these "best
decisions" (relative to various assumptions) will also be
multiplied by .
An information value being the variation of a function in the vicinity
of an extremum, the information values will rather be multiplied by
. Therefore, the phenomenon described above will therefore
be amplified if the dispersion increase, but the relative orders of
magnitude will not change.
(Scarf's rule)
