4 How to summarize the demand dispersion ?

In the former Section, we have assumed that and were precisely known. In 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 in the specific context of optimizing the newsboy problem.

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

4.1 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. By the definition of the mean, we have . Substituting the variable in eq:cost-of-uncertainty by this expression of , we obtain:

This formula has the same appearance as the cost of uncertainty formula eq:ckoi-cost. But, now, the parameters , and are relative to the mean. Thus the quantity defined by:

is independent of the cost to price ratio and depends only from the distribution . By the very definition of , we have the bound:

The quantity (further referred as the intermeans parameter) has some similarity with the interquartile range, where one subtracts 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 subtract the mean of the left part from the mean of the right part, both parts being separated by the mean of the population, the result being afterwards multiplied by . Therefore appears to be a measure of the dispersion of the distribution.

Since this use of 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 their skewness.

Tab. 1: Comparison between the intermeans 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. Moreover, it can be noticed that so that:

4.2 An upper bound for the cost of uncertainties

From eq:cost-of-mean and tab:comparison-delta-sigma we have the following result: if the demand is assumed to be either normal or lognormal, the cost of uncertainties is bounded by:

This bound depends only on the selling price (and the dispersion), but doesn't depends on the cost to price ratio. In the general case, the Scarf theorem eq:scarf-bound only ensures that:

4.3 New graphical proof of the Scarf's bound

Many alternative proofs of the "Scarf's rule" have been given ([Gallego 1993,Yue 2006]). What follows concerns the second part of the original proof. Apart its relevance to the Scarf's theorem, this proof will indicate how to find the best decision against the family.

The proof given by H. Scarf for the bound eq:scarf-bound was divided in two steps, the first one being to prove that, over all the distributions , the worst case for a given is obtained by an ad hoc "two Dirac's" distribution. Therefore finding the best order against is reduced into finding the best order against .

The second step can be done as summarized by fig:maxminscarf (where , , and , inducing ). All the curves corresponding to the different values of the ordered quantity are going through the same point, whose abscissa is . More precisely, all these curves are made of rectilinear and "parabolic" pieces and all the complete "parabolae" are going through this same point. The conclusion is straightforward: the best decision is the whose curve admits this special point as its minimum.

Fig. 4: Searching the max-min over the two Dirac's distributions.

[Knowing

.]

[Knowing

.]

Having seen the fact of the confluence, a formal proof is easy to obtain. The "parabolic" part of can be rewritten as:

and eq:scarf-bound follows (according to whether is or not an acceptable value for given the value of ).

4.4 Applying the same method when are known

When using two Dirac's distributions, we have so that , the greatest value corresponding to , i.e. the symmetric case. Therefore, the assumptions "knowing " and "knowing " are no more quite equivalent as they were in tab:comparison-delta-sigma.

fig:maxmindou show what happens when drawing the curves for different values of the ordered quantity . Now, the "parabolic" parts are straight lines. But here again, all of them are going through the same point whose abscissa is This is comforted by:

And therefore, the robust decision is no more given by eq:scarf-bound but by:

In other words: if you consider that the inter-means parameter is the right way to summarize your knowledge of the dispersion of the demand, then the robust order against all the two Dirac's distributions is nothing but the mean (unless the product is so few profitable that doing nothing becomes better).