Let us assume now that only 
and 
are known and examine
what can be said when 
ranges over all the elements of a family
of models that all fit these characteristics. In such a case, we can
determine a robust value 
for the order quantity by
the following algorithm : for each value of 
, we determine
the worst distribution of the family, i.e. the that minimize
the expected gain. And we choose the order quantity that optimize
the worst case. In other words, we solve:
The quantity 
is the worst possible expected revenue induced by any pdf having the
given characteristics 
. The obtained order
decision is robust in the sense that any possible pdf in the family
will give
at least the gain 
, representing the guaranteed expected
performance. For normal or lognormal family,
reduces to only one possibility, leading to the same result as in
Section 1.3. It is of interest to study
various families of models with a third or more free parameters.
This is the case for triangular models.
When the family of models is the set of all triangular distribution
having given values for 
, one degree of freedom
remains : the shape 
. When 
are
known, the values of 
(resp. the minimum
possible, the most probable and the maximum possible demand) are given
by:
should be positive. It can be seen that if 
,
then all values of 
are allowed,
while 
allows only 
where 
is the useful solution of an equation of second
degree. As a result, the set of the effective values for the couple
is the grayed zone in FIG. 5(a).
In this new situation, the method to find the best decision is exemplified
in the rest of FIG. 1.5, where we have taken
, 
and 
(leading to 
).
In each sub-figure, there are several curves, each one labeled with
a value of . For example, the curve labeled ''
''
describes what is the expectation of the gain, knowing that 
,
but depending on the value of . In other words, this curve
is the graph of the function 
.
For each curve, the worst case is marked by a circle. It can be seen,
in FIG. 5(b) and FIG. 5(c),
i.e. for 
and 
, that the worst case
ever occurs when 
. This can be confirmed by formal computation.
A more precise result is:
This result is to be compared with the following : in 1958, H. Scarf
has proven that, over all distributions having the given characteristics
, the worst case for a given is attained
by a "two Diracs" model. Therefore the solution
of the problem
. Notice
that in this case the robust solution 
is not equal to
unless 
or 
. As expected, the
value of obtained by eq:scarf-bound is
less than the obtained when considering only a smaller
family of models. The solution is then a conservative
result.
|
[Possible associations for
 .]
[Assuming
.]
[Assuming
.]
|
