We have mentioned all these conceptual difficulties in order to emphasize the fact that the probability model, by it's very nature, may assume more knowledge that we can ever hope to obtain. Therefore, it must be carefully checked if conclusions obtained using a given model are robust relative to a change towards another model that remains compatible with our actual knowledge.
In what follows, we will examine what happens, for several distributions,
when the variance increase (using 
).
Let us start by the normal law since this distribution is widely used to illustrate topics related with inventory problems. Indeed, using this law to model the demand would be unquestionable if consumers were processes having independent behaviors. But this is quite never the case. Many decision criteria are the same for everybody, the welfare of the general economy among them.
Moreover, using the normal law to describe a positive random variable
necessitates a small 
in order to ensure that 
.
fig:normal-pdf shows that 
is un upper limit
of what can be used, while a numerical computation shows that 
when 
. Using greater values for would be unrealistic.
In the two other pictures of fig: normal, we have plotted
the values of the expected gain 
against the corresponding
decision 
. More precisely, fig:normal-g shows what
happens when 
and fig:normal-d is related
to 
. In both cases, we have a broken line corresponding
to 
, and three curves, the lowest being relative to the
greatest .
The small line starting from 
is the
locus of the extremal points. Let us write 
and 
for the
pdf and the cdf of the reduced normal law, and define 
by 
.
Then elementary computations based on eq:Q-optimal and
eq:G-value lead to:
,
the locus of the extremal points is a line segment.
When modeling a positive quantity, the lognormal law is obviously a better candidate than the normal law since the lognormal distribution doesn't introduce artificial negative values. Nevertheless, it should be kept in mind that using this model is roughly equivalent to assume that the solvable demand is the product of many independent random positive factors. Clearly, this is not ever the case.
fig: log describes what happens when using this model.
fig:log-pdf shows the pdf's corresponding to ,
while the other two are plotting the expected gain
against the corresponding decision in the four cases 
.
As before, fig:log-g assumes and fig:log-d
assumes .
Now, 
can grow to infinity, inducing the existence of a
``fat tail'' for the distribution. Thus, quite all the mass should
concentrate towards 0 in order to equilibrate the ``fat tail''
since the mean has to remain constant. This is the reason
why 
when 
, as it can be
seen on both graphs. But while, in fig:log-g, the locus
shifts ever to the left, we can see, in fig:log-d, that
the locus starts shifting to the right (due to the value of 
)
and afterwards turns to the left (and tends to the origin).
As said in sec:Discussion-about-hypotheses, it is not
realistic to assume that the distribution of the demand is exactly
known. We can at best extract some knowledge from the collected historical
data so that actual problems are rather ``fuzzy problems''. But,
when using one of the former models, the parameters and
are the only degrees of freedom available.
Therefore it is of interest to use simple models, but depending of
at least three parameters, to test how robust are the conclusions
drawn from partial information. Let us call triangular distribution
a model whose pdf looks like fig:trian-pdf. If we note
by 
, 
and 
, respectively, the 
mode and 
of the distribution, the function 
is given
by:
as great 
.
In comparison, it has been seen that 
for the normal distribution.
The triangular model is is a simple way to deal with the fact that
often the demand is not symmetrical around it's mean. This lack of
symmetry is usually measured by the 
,
where 
is the third centered moment. This moment has a nice
expression over 
:
(the
mean), 
(the width) and 
(the barycentric position of in 
,
verifying 
) to characterize the distribution.
We obtain:
. Conversely, the
skewness gives the shape (i.e. ), then gives the
size (i.e. 
) and finally fixes the position of the
triangle along the axis. By this method, one can deal with skewness
up to 
.
fig: trian has been drawn using 
(i.e. assuming
that is exactly known). In fig:trian-g, the
skewness of the distribution and the cost to price ratio are acting
in conjunction, and the locus of the extremal points shifts clearly
to the left. In fig:trian-d, these two factors are acting
in opposition, and the shift to the right of the corresponding locus
is not so strong.
Another model with three parameters is the ``two Dirac's model''
that has been introduced by Scarf [4] to obtain his max-min
formula. In this model, the parameters 
are defined by:
 |
 |
 |
|
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|
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These formulae can be inverted, and we obtain:
should remain positive and, as increases,
the range 
of the allowed values for
shortens. With some computations, we obtain fig: scarf.
when 
).
