The supply chain planning requires the definition of the ordered quantity and the inventory level which optimize profit. For companies which cannot build distribution processes with a make-to-order strategy, these decisions depend on the knowledge of the future demand which is mainly expressed by a probability distribution function (pdf). In many papers, for example those related to the well known newsboy problem [8], this pdf is assumed to be exactly defined. In that case, the optimal order quantity has a simple and elegant expression which can be extended to various situations such as multi products environment with multiple discounts [6], multi period sales and ordering, capacity constraints [11], or random procurement delays.
However, this assumption can hardly be verified since the demand forecasting model cannot be easily validated especially for turbulent and wildly changing markets: high competition between companies, customer volatility, unpredictable fashion trends, high degree of diversity, and short product life cycle. This situation mainly has three consequences [1].
Firstly, it is difficult to conduct a real experiment for the verification of the demand models. This experiment would need a repeatable context. Secondly, the complexity of the market or the economic context may lead to the impossibility to verify various modeling hypotheses such as ergodicity, independence, ...
Lastly, especially for make-to-stock industry and commerce, it is also impossible to quantify the actual demand even afterwards. One can only observe the satisfied demand materialized by a receipt or an invoice, or observe that a stock-out has occurred. When the inventory becomes empty, a common situation is that one cannot guess anything about the unobserved demand.
The impossibility to identify exactly this demand distribution function from the actual knowledge requires more robust optimization methods which are addressed by a min-max search in the expected gain.
In the min-max optimization, the primary hypothesis on the knowledge
of the demand probability distribution function is reduced to the
knowledge of fewer parameters that define a family of demand models.
As a prerequisite, we assume that the mean demand is reasonably detected
and identified by the statistical analysis of past sales data. Then,
the min-max method looks for the decision which maximizes the minimum
given criterion for a family of models 
, defining the ``least
bad'' decisions. Following the method introduced by [8],
many papers [4,12] have studied the
optimal order for the family 
of pdfs that share
the same mean 
standard deviation 
.
Recently, from the consideration of a new expression of the cost of
uncertainties, we have introduced [2] a new measure
of the dispersion, the intermeans parameter 
(cf. Eq. 7
and Section 4 in the present paper). And we have shown that the optimal
decision differs from "the Scarf's rule" when the
min-max method is applied to the family 
of
pdfs that share the same mean and intermeans parameter .
In this context, we have undertaken a comparative study of the properties
of both this new measure of the dispersion (
and the usual
one 
. Among the properties investigated, the
sampling distribution of has been studied, in order to provide
some confidence levels from limited historical sales data.
Closed form expressions have been obtained for the "two Dirac's"
and uniform distributions. For the general case, we have conducted
numerical simulations in order to compare the behaviors of the estimators
of both and . Interesting directions for future
research have been obtained.