3 Max-min problems

3.1 The Scarf's rule

Let us assume now that only and are known and examine what can be said when ranges over the family of all the distributions that share these parameters. In such a case, we can determine a robust value for the order quantity by the following algorithm : for each value of , we determine the worst distribution of the family, i.e. the that minimize the expected gain. And we chose the that optimize the worst case. In other words, we solve:

In 1958, H. Scarf has proven that, over all distributions sharing given values of , the worse case for a given is ever a "two Dirac's" distribution. Therefore the best decision against the whole family can be obtained by taking into account only these "two Dirac's" distribution. Assuming that , the solution, as published in [4], is given by:

3.2 If, nevertheless, we 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. 2 by this expression of , we obtain:

This formula has the same appearance as the cost of uncertainty formula Eq. 3. 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 on the distribution . By the very definition of , Eq. 5 leads to the following bound for the cost of uncertainties:

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.

3.3 Some properties of the intermeans parameter

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 Table 1, 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
triangular	1/
lognormal
"two Dirac's"

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:

3.4 An upper bound for the cost of uncertainties

From Eq. 7 and Table 1 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. 4 only ensures that:

3.5 Robust decisions against the "two Dirac's"

Many alternative proofs of the "Scarf's rule" have been given ([2,6]). The proof given by H. Scarf for the theorem Eq. 4 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 Figure 1(a) (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 order whose curve admits this special point as its minimum.

Fig. 1: Searching for the max-min over the two Dirac's distributions.

[Knowing

.]

[Knowing

.]

When using the first three lines of Table 1, the assumptions "knowing " and "knowing " are quite equivalent. On the contrary, when using the two Dirac's distributions, we have so that ranges over the whole interval and these assumptions become strongly different.

Figure 1(b) 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 confirmed by:

And therefore, the robust decision is no more given by Eq. 4 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).