Let us assume now that only 
and 
are known and examine
what can be said when 
ranges over all the elements of a
``bucket of models'' that all fit these parameters. 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 bucket, i.e. the that minimize
the expected gain. And we chose the that optimize the worst case.
In other words, we solve:
When the ``bucket of models'' is the set of all triangular distribution
having fixed values for 
, one degree of freedom
remains : the shape 
. When 
are
known, the values of 
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:tau-cv.
In this new situation, the method to find the best decision is exemplified
in the rest of fig: maximin-trian, where we have taken
, 
and 
(leading to 
).
In each subfigure, 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:maxi-trian-g and fig:maxi-trian-d,
i.e. for 
and 
, that the worst case
ever occurs when 
. This can be confirmed by formal computation.
And therefore we have the following result : against all triangular
models (assuming 
), the decision 
is the more
robust.
The former result is to be compared with the following. In 1958, H.
Scarf has proven that, over all distributions having given
, the worse case for a given is ever a ``two
Dirac's model''. Therefore the solution of the problem
obtained by eq:scarf-bound
is less than the obtained when considering only a smaller
``bucket of models''.
.]
