4 How to summarize our knowledge ?

4.1 Discussion about hypotheses

Let us now discuss about the meaning and the potential validity of the hypotheses of the preceeding sections. The first point to be mentioned concerns what is modeled. Using a probability distribution , we can model our lack of actual knowledge concerning the future demand. We may for example think that a better knowledge (at ordering time) can be reached but that its cost would be far beyond the additional benefits resulting from this additional knowledge. Another point of view is that markets are intrinsically wild so that the probability function rather models the very nature of market [2]. This point, of philosophical nature, is obviously an open question, but nevertheless a key point.

The second point concerns what experimental procedure can be used to determine . A "gedachte Experiment" is as follows : starting with a great number of exact copies of the actual world, put different orders in these worlds, inducing them to evolve (independently) in different manners and observe what happens at selling time. One cannot escape this point of view by considering approximations obtained from times series, since only ergodicity can justify such approximations (without mentioning the fact that actual times series are quite ever too short to conclude, even assuming ergodicity).

A third point is that the "actual" demand cannot be measured (even afterwards) when the demand overflows the inventory. In such a case, the only actual knowledge is the fact that . Therefore, a slight shift towards overaging could be a good policy since it results into a better knowledge (for a slight cost).

We have mentioned all these conceptual difficulties in order to emphasize the fact that the probability model, by it's very nature, may assume more knowledge that we 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.

In this paper, we have assumed that and are precisely known. In the practice, we have to guess them from what is really known. The fact that these parameters are the ones usually used to summarize the dispersion properties of a population doesn't prove that these parameters are the optimal ones for the specific context of optimizing the newsboy problem.

To introduce what other parameter could be used to evaluate the dispersion of the demand, let us consider in detail what happens when the ordered quantity is chosen exactly equal to the expected demand.

4.2 If we nevertheless buy the average demand ?

Knowing that is not the optimal inventory, we can nevertheless examine what happens when this choice is taken and define . By the definition of the mean, we have . Substituting the in eq:cost-of-uncertainty by this expression of , we obtain :

In this formula, , and are relative to the mean, i.e. is the average of the that exceed etc. This formula gives an upper bound for the cost of uncertainty. Its interest comes from the fact that this bound is expressed as the product of a price (the selling price of an unit) and a number

that appears to be a measure of the dispersion of the demand.

This has some similarity with the interquartile range, where one substracts the median of the left hand part from the median of the right hand part, both parts being separated by the median of the population. Here we use the means and substract the left part's mean from the right part's mean, both parts being separated by the population's mean, the result being afterwards multiplied by .

Since the idea to use this quantity as a measure of the dispersion seems to be new, we have undertaken a comparison between this quantity and the usual measure of the dispersion, i.e. the standard deviation . This leads to tab:comparison-delta-sigma, where it could be noticed that the ratio is quite the same for all triangular distributions, whatever shape they have.

Table 1: Comparison between the new parameter and the usual .

.

distribution	(exact)	(approx)
uniform
normal
lognormal
triangular	1/

For the lognormal distribution, the ratio varies. For the wildest ones, i.e. for , one has . But, in the not so wild case, i.e. when the demand has only one chance over thousand to go outside the interval , the relation holds.

4.3 Asserting a result

From tab:comparison-delta-sigma and the obvious fact that , we have the following result : if the demand is assumed to be Gaussian, the cost of uncertainties has a simple bound, namely :

This bound depends only on the selling price (and the dispersion), but doesn't depends on the cost to price ratio.

4.4 The Scarf's bound

H. Scarf has proven in [1] that, over all distributions having given , the worse case for a given can be obtain by an ad hoc two-points distribution. Therefore the solution of the problem

can be obtained by taking into account only the two-points distributions. The solution obtained is as follows :

For these two-points distributions, we have so that , the greatest value corresponding to , i.e. the symmetric case. Therefore, it is not clear if the standard deviation is really the most efficient parameter to summarize the dispersion of the demand. The measure could be a better statistic.