3 Information value of assumptions

Let us now discuss the meaning and the potential validity of the hypotheses of the previous Section. Such a discussion is obviously the key point when trying to induce a practitioner to adopt any academic formula to conduct his own business.

3.1 About the Meaning of the Distribution $\Phi$

The first point to be mentioned concerns what is modeled. When using a probability distribution $\Phi \left(\xi \right)$, 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 greater than 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 (Corker et al., 1986).

The second point concerns what experimental procedure can be used to determine $\Phi \left(\xi \right)$. 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).

Tab. 2: Best decisions, depending on the assumed knowledge.

assumed $\Phi$ or $\mathcal{F}$	$${\frac{c}{r}}=.3$$	$y^{*}$	$G$	$${\frac{c}{r}}=.7$$	$y^{*}$	$G$
$\Phi \left(\mu , \sigma, normal\right)$		$1108$	$5710$		$807$	$1878$
$\Phi \left(\mu , \sigma, lognormal\right)$		$1117$	$5916$		$821$	$2069$
$\Phi \left(\mu , \sigma, best shaped triangle\right)$		$1109$	$5654$		$788$	$1892$
$\mathcal{F}\left(\mu , \sigma, triangular\right)$		$1103$	$5607$		$813$	$1774$
$\mathcal{F}\left(\mu , \sigma\right)$ (Scarf's rule)		$1083$	$5393$		$833$	$1560$
$\mathcal{F}\left(\mu , \delta , Dirac\right)$		$958$	$5541$		$958$	$1709$
$\mathcal{F}\left(\mu , \delta , triangular\right)$		$1159$	$5807$		$818$	$1807$

3.2 About the Not Observed Demand

A third point is that the actual demand cannot be observed (i.e. measured, even afterwards) when the demand overflows the inventory. In such a case, the only actual knowledge is that an overflow has occurred. Obtaining any quantity concerning the demand (i.e. the whole range $\left[0, \infty\right]$ of the $\Phi$ distribution) only from what happens in the range $\xi \in\left[0, y\right]$ (i.e. from only the observed demand) cannot be done in a 'distribution free' manner. Even the mean $\mu$ cannot be guessed that way.

This fact is a key point when discussing the meaning of the Scarf's theorem. In the original paper (Scarf, 1958), the author only states: "Let $\mu , \sigma$ be fixed. Then...". Afterwards, other authors have presented this theorem as a 'distribution free' result, on the basis that the "Scarf's rule" is issued from a family of distributions rather than from a given distribution. But, in our opinion, this presentation is too optimistic.

Moreover, any 'educated guess' of the $\xi >y$ part of the reality cannot be 'cost free' either. And this cost must be incorporated into the total income we want to optimize. For example, if the newsboy increases his order quantity to observe more demand, he will certainly increases his knowledge of the demand, but the core question was increasing his income.

3.3 Numerical Example

Let us now take an example and examine what can be done when historical data are available. For example, if the last $n=16$ orders were $y=999$ and the actual past sales were:

$$\begin{array}{c} 743, 999, 999, 999, 851, 939, 601, 483\\ 999, 655, 999, 856, 821, 810, 999, 999\end{array}$$

then we have the following estimators:

$$\mathrm{est}\left(\theta _{999}\right)=0.44\quad;\quad\mathrm{est}\left(\xi ^{o}_{999}\right)=751$$

But, as discussed before, nothing can be said on the really interesting parameters, namely $\mu , \sigma, \delta , \xi ^{o}=\xi ^{o}_{\mu }, \xi ^{u}$. To estimate these parameters, we must recreate the missing data, i.e. the exact values reached by the demand when it has overflowed the inventory. This can be done in many ways but, in any case, the figures obtained are only fictional, and have an extra cost. Let us assume the following non fully observed demands:

$$\begin{array}{c} , 1278, 1418, 1461, ., ., ., .\\ 1176, ., 1016, ., ., ., 1194, 1028\end{array}$$

leading to the following estimators:

$$\begin{array}{c} \mathrm{est}\left(\mu \right)=958\;;\;\math... ...\right)=287\;;\;\mathrm{est}\left(\delta \right)=116\end{array}$$

In Tab. 2, these estimators have been used to compute the best decisions against three distributions and four families. The values $r=10$ and either $c/r=0.3$ (left part) or $c/r=0.7$ (right part, not otherwise commented) have been used. The first five lines describe what are the consequences of "assuming an exact knowledge of $\mu , \sigma$", while the last two are relative to the consequences of "assuming an exact knowledge of $\mu , \delta$".

If nothing else than $\left(\mu , \sigma\right)$ is assumed, the Scarf's robust decision ensures a expected gain of $5393$. The additional value of an information concerning the shape of $\Phi$ is given in Tab. . It should be noticed that these values are not obtained by subtracting the corresponding gains from Tab. 2 (when you are not aware that $\Phi$ has a given shape then, nevertheless, you are not playing the "Scarf's rule" against the worst "two Dirac's" distribution, but against this given $\Phi$). It can be seen that, in our example, the value of these informations relative to the shape is negligible.

In Tab. 3, the value of an information relative to which is the right measure of the dispersion is computed (assuming that the play is against the "two Dirac's" distributions). It should be noticed that these values are significantly greater than those of Tab. .

Tab. 3: Information values concerning how to summarize the dispersion.

$$\begin{array}{c} \mathrm{Factual}\:\mathrm{shape}\\ \mathrm{of}\:\mathcal{F}\end{array}$$	$$\begin{array}{c} \mathrm{Newsboy's}\\ \mathrm{belief}\end{array}$$	$$\begin{array}{c} \mathrm{best}\:y\:\mathrm{for}\:\mathrm{this}\\ \mathrm{belief}\end{array}$$	$$\begin{array}{c} \mathrm{worst}\:\theta \:\mathrm{against}\\ \mathrm{this}\:\mathrm{best}\:y\end{array}$$	$G\left(y, \Phi \right)$	$$\begin{array}{c} \mathrm{value}\:\mathrm{of}\\ \mathrm{information}\end{array}$$
$\mathcal{F}\left(\mu, \sigma, Dirac\right)$	$\delta$ constant	$958$	$\theta =0.5$	$5273$
$\mathcal{F}\left(\mu, \sigma, Dirac\right)$	$\sigma$ constant	$1083$	$\theta =0.3$	$5393$	$120$
$\mathcal{F}\left(\mu , \delta , Dirac\right)$	$\sigma$ constant	$1083$	$\theta =0$	$5389$
$\mathcal{F}\left(\mu , \delta , Dirac\right)$	$\delta$ constant	$958$	$all$	$5541$	$152$