2 The newsboy problem

2.1 Description of the newsboy problem

Let us consider what is known as "the newsboy problem" and use the notations of [1]. We have the opportunity to purchase now an amount of some good, at unitary cost (regardless of the quantity purchased). It is assumed that the future demand distribution is precisely known, that the future unit selling price is known and independent of the number sold and that non sold units are discarded.

In this Section, we will examine what can be inferred there from and postpone the discussion of the validity of such hypotheses to sec:Discussion-about-hypotheses.

The satisfied demand will be , leading to the actual gain : . The usual criterion used to fix the optimal quantity to buy is maximizing the expected value of this gain. Denoting this expectation by , we have :

Derivating, one obtains the condition , i.e.

In other words, the best quantity you can buy is not the expectation of the demand.

2.2 The cost of uncertainty

Let us now compare the eventual gain with the naive value , i.e. with the gain that will occur if we buy right now the expectation of the future demand and if, by chance, it happens that we effectively sell these units. Defining as the probability of a too small order and (resp. ) as the expected value of the demand knowing that (resp. knowing that ), we have :

A straightforward computation leads to :

Since the right hand side is obviously positive, is the Holy Graal of the problem. Without uncertainties you can choose so that will be the expectation of your gain (the solution being . But in presence of uncertainties, there is no choice of that allows you to expect a gain reaching this value.

Moreover, this quantity has a clear meaning in terms of risks evaluation. You have a risk that the demand overflows your inventory . And in this case, your score is burdened by the fact that you miss the opportunity to sell units, leading to an average miss to gain of . On the other hand, you have a risk that your inventory overflows the demand. And in that case, your score is burdened by the resulting leftover units, leading to an average extra cost of .

Therefore, the right hand side of eq:cost-of-uncertainty is nothing but the cost of uncertainty. The best choice for can decrease this cost, but it never vanishes.