3 Behavior of various families of distribution functions

As already stated in the introductory Section, assuming the knowledge of the probability distribution function of the future demand may assume more knowledge than one can ever hope to obtain. Therefore, it must be carefully checked if conclusions obtained using a given model are robust relative to a change towards another model that remains compatible with our actual knowledge. These questions will be more precisely examined in the later Section 1.4.

In the present Section, we will assume that the mean of the demand has been identified and we shall examine how the optimal order decision changes when the variance of the demand or the assumed shape of the pdf are changing. We will consider the usually used normal model, as well as the lognormal, triangular and Scarf's "two Diracs" models. In order to illustrate phenomena resulting from the increase of demand variance, we will consider the numerical values .

1 Normal model

To illustrate the various topics related to inventory problems, let us start with the normal probability law since this distribution is widely used. Using this Gaussian model for the demand process would be unquestionable if consumers were (additive) processes having independent behaviors. But in the context of textile-apparel or more generally for many consumer goods, this is quite never the case since many buying decision criteria are the same for everybody (current fashion trends, common factors from national or global events, the welfare of the general economy ...), so that buying decisions are likely to be correlated, invalidating the Gaussian central limit theorem hypotheses.

Moreover, this model allows negative values for the demand , an assumption that is unrealistic except for rare cases. Eliminating the negative values through a sufficiently small probability threshold requires that remains small. For example, requires that . In FIG. 1(a), this limitation is apparent for the lower larger curve (): using greater values for would usually be unrealistic.

In the two other pictures of FIG. 1.1, we plot the values of the expected gain versus the corresponding decision for various uncertainty level. With the same average demand , a greater uncertainty modeled by a greater leads to a lower expected gain. More precisely, FIG. 1(b) shows what happens when % and FIG. 1(c) is related to %. In both cases, we have a -shaped broken line corresponding to , and three curves, the lowest being relative to the greatest .

The small line starting from is the locus of the extremal points . Let us denote and for the pdf and the cdf of the reduced normal law, and define by . Then elementary computations based on eq:G-value and eq:Q-optimal lead to:

Therefore, when the distribution is normal with a given mean , the locus of the extremal points is a line segment which tends to the left, (resp. to the right, ) when the cost ratio verifies % (resp. %).

2 Lognormal model

When modeling a positive quantity, the lognormal law is obviously a better candidate than the normal law since the lognormal distribution does not introduce artificial negative values. Nevertheless, it should be kept in mind that using this model is roughly equivalent to assuming that the solvable demand is the product of many independent random positive factors, like the size of the population, the welfare of the economy, etc. Clearly, this is hardly the case.

FIG. 1.2 describes what happens when using this model. FIG. 2(a) shows the pdf's corresponding to , while the other two figures are plotting the expected gain versus the corresponding decision in the four cases . As before, FIG. 2(b) assumes and FIG. 2(c) assumes .

Now, the standard deviation can grow to infinity, inducing the existence of a ''fat tail'' for the distribution. Thus, as shown in FIG. 2(a), most of the mass should concentrate towards in order to equilibrate the ''fat tail'' since the mean has to remain constant. For this reason, we have when , as it can be seen on both other graphs. But while, in FIG. 2(b) where , the value of always decreases when increases, we can see in FIG. 2(c) that increasing from induces in a first time an increase of from (due to the value of ) followed by a decrease towards zero when becomes bigger and bigger.

3 Triangular model

As said before, it is not realistic to assume that the distribution of the demand is exactly known. This will be further discussed in Section 1.4. We can at best extract some knowledge from the collected historical data so that actual problems are rather ''fuzzy problems''. When using one of the former models, the parameters and are the only available degrees of freedom.

Therefore it is of interest to use a simple model, but nevertheless depending on at least three parameters, to test how robust are the conclusions drawn from our limited knowledge. Let us call triangular distribution a model whose pdf looks like FIG. 3(a). If we note by , and , respectively, the mode and of the distribution, the function is given by:

while mean and variance are:

These formulae show that the upper limit of the coefficient of variation of a triangular distribution is . In comparison, it has been seen that is an acceptable limit for the normal distribution, while the lognormal model allows .

The triangular model is a simple way to deal with the fact that often the demand probability function is not symmetrical around its mean. This lack of symmetry is usually measured by the , where is the third centered moment. This moment has a nice expression over :

But we can obtain a more compact expression to characterize the distribution by using : the mean, : the width, and : the barycentric position of in , with . We obtain:

and therefore the skewness depends only on . Conversely, the skewness gives the shape (i.e. ), then gives the size (i.e. ) and finally fixes the position of the triangle along the horizontal axis. With this method, one can deal with skewness up to (value obtained when ).

FIG. 1.3 has been drawn using (i.e. assuming that is exactly known). In FIG. 3(b), the skewness of the distribution and the cost to price ratio are acting in conjunction, and the locus of the extremal points shifts clearly to the origin . In FIG. 3(c), these two factors are acting in opposition, and the shift to the right of the corresponding locus is not so strong.

4 Two Diracs model

Another model with three parameters is the "two Diracs" model that has been introduced by Scarf (1958) to obtain his max-min formula. In this model, the pdf is reduced to only two possible demand quantities or .

The parameters are defined by:

while straightforward computation gives :

Obviously, should remain positive and, as increases, the range of the allowed values for shortens.

With some computations, we obtain FIG. 1.4. The locus of the optimal order quantity is on the envelop of the maximal gain when varies, while is either (when ) or (when ).