One of the most common problem in the area of the supply chain management is to take decisions concerning inventory and delivery questions in order to obtain the maximal possible profit. For that, the key source of information is a forecast of the demand, that is often based on sales historical data.
One of the first paradigm used to study this problem is the newsboy model, which requires a knowledge of the stochastic distribution of the demand. Several variants have been developed, using multi-period, multi-products, presence of extra costs : obsolescence, salvation, etc. [Tang 2002,Groenevelt 2003]. This problem has a very simple and elegant analytical solution if one assumes an exact knowledge of the distribution of the demand. Nevertheless, when applying this solution, the Gaussian distribution is often used, sometimes without justification, or outside its natural range of validity.
The aim of the present paper is to analyze the consequences of the limited knowledge we have upon the distribution of the future demand. This question is usually addressed by a max-min search in the expected gain. Another possible approach, namely a min-max search in the expected ex post regret can also be used [Vairaktarakis 2000,Perakis 2006], but this will not be investigated here.
We will focus our attention onto the simplest newsboy model and consider
how the dispersion of the population can be summarized during the
max-min search. When the usual family 
is
considered, the Scarf's theorem follows [Scarf 1958,Gallego 1993,Yue 2006].
In this paper, we introduce another measure of the dispersion, namely
the inter-means parameter 
eq:ckoi-definition,
and it will be proven that assuming an exact knowledge of 
instead of 
leads to a different conclusion.
Our paper is organized as follows. In sec:The-newsboy-problem
we restate the hypotheses and notations usually used [Scarf 1958]
and recall the elementary formulae for the best order quantity and
the associated gain. The usual formula giving the cost of uncertainties
is recalled and also rewritten in a way that introduces the parameter
. The behavior relative to usual distributions (normal, lognormal
and "two Dirac's") is briefly recalled. The family
of the triangular distributions is introduced in order to take the
skewness into account while keeping realistic.
In sec:Max-min-problems, the uncertainty over the distribution
is addressed by the usual max-min method. For each order quantity
, the worst demand distribution among a given family is determined,
and thereafter the value of is chosen in order to maximize
the "gain in the worst case". The best decision
against the family 
is determined
and compared with the general Scarf's solution, i.e. with the best
decision against 
.
In sec:How-to-summarize, we analyze how to describe the
dispersion of the demand and introduce a new measure of this dispersion,
namely the intermeans parameter . A new proof of the
"Scarf's rule" is given. Reusing the underlying
idea of this proof, it will be proven that, against the family 
(i.e. the "two Dirac's" distributions sharing the
same 
), the robust decision is 
and therefore
significantly differs from the robust decision adapted to .
In sec:Discussion-about-hypotheses we discuss the hypotheses
that were at the basis of the preceeding results. It will be shown
that obtaining or
is neither 'distribution free' nor 'cost free'. A careful examination
of what kind of experiments can be done to catch these parameters
is undertaken, and a numerical example is proceeded.
The paper ends by some concluding remarks and a bibliography.