Let us assume now that only 
and 
are known and examine
what can be said when 
ranges over the family 
of all the distributions that share 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 family, i.e. the that minimize
the expected gain. And we chose the that optimize the worst case.
In other words, we solve:
When the family is the set 
of
all triangular distributions having fixed values for 
,
one degree of freedom remains : the shape 
. When 
are known, the values of 
are given by:
should remain 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 illustrated
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.
Assuming that 
is allowed (i.e. ),
we have the following result:
Some computations lead to 
in the central case, to 
for small 
and to 
in the last case. In fig:maxi-trian-g we have 
and in fig:maxi-trian-d 
.
The former result is to be compared with the following. In 1958, H.
Scarf has proven that, over all distributions sharing given
values of , the worse case for a given is ever
a "two Dirac's" distribution. Therefore the best
decision against the whole family can be
obtained by taking into account only these Scarf's distribution, and
the solution, as published in [Scarf 1958], is given by:
obtained by eq:scarf-bound
is less than the obtained when considering only a smaller
family: here, we have now 
while the same values
of the parameters were leading to 
in the only
triangular case.
