2 The newsboy paradigm

1 Description of the problem

The paradigm known as "the newsboy problem" involves two time periods: a time for product purchasing decision, and a time for the future sales when the result of the previous decision can be established. Let us suppose that we have the opportunity of ordering now a quantity of some good at unitary cost . Let us additionally suppose that we know what is the probability distribution of the future demand , as well as the exact unitary price at which the sales will occur. At the end of the sales period, non sold products are supposed to be discarded. Without loss of generality, the common case of non zero discount price for sales occurring at will not be studied. We also suppose that no additional quantity can be purchased later than . Then, we have no opportunity to correct any forecasting error.

As usual, we denote the probability density, and the cumulative distribution. The satisfied demand will be if it happens that the available products exceed the demand quantity and otherwise, leading to the gain where . The usual criterion for the identification of the optimal value of the ordered quantity is maximizing the expected value of this gain. Denoting this quantity by , we have:

The derivation of 1.1 leads to the condition , i.e.

The best order quantity is associated with the critical fractile and therefore usually differs from the expectation of the demand . The value is smaller when the cost ratio is higher: cheap and profitable products can be purchased at greater quantity without increasing the financial risk whereas the risk is obviously greater on expensive products. Moreover, it is clear that when assuming .

2 The cost of uncertainty

The uncertainty on the demand introduces a diminution of the expected revenue that can be defined as the cost of uncertainty: a loss of profit due to stock-out or an extra cost due to discarded inventory. Let us compare the eventual gain with the naive value , i.e. with the gain that will occur if the retailer purchases the expectation of the future demand and if, by chance, it happens that we effectively sell these units. Defining as the probability that and (above) --resp. (below)-- 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 Grail of the problem, the maximum gain in the ideal case. Without uncertainties, the value of can be chosen in order to obtain for the maximized gain (the solution being . But in presence of uncertainties, the value becomes unreachable.

Moreover, the difference has a clear meaning in terms of risks evaluation. A retailer has a risk that the demand overflows his inventory . And in this case, his score is burdened by the fact that he misses the opportunity to sell units, leading to an average loss of earnings of . On the other hand, the retailer has a risk that his inventory exceeds the actual demand. And in that case, his 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 the cost of uncertainty. A better choice for can decrease this cost, but it never vanishes. Equation eq:cost-of-uncertainty can also be used to investigate the gain obtained from reducing uncertainties, as illustrated in FIG. 1.1, 1.2, 1.3 and 1.4.