3 MIN-MAX AND NEWSBOY MODEL

The purchasing problem is stated as follows: assuming a given knowledge of the future demand, the decision maker has to buy now a certain quantity of products in order to maximize his revenue during a future sales period. Because of the short life-cycle of most of the textile products, we suppose that there is no opportunity to correct any error that he may have made on this order quantity. As an illustration, the procurement delay for a fabric may be about three months, while the logistic and transportation delay may range from one to three weeks or more for relocated industries. In that case, the purchase decision should be taken about four months before any sale. Any error cannot be corrected if this sales period is of three months (one season). In order to get a higher comprehension of the phenomena, we will simplify the problem and illustrate it by the very simple newsboy model.

3.1 Description of the newsboy problem

Let us consider what is known as "the newsboy problem" and use the notations of [8]. 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 exactly known by its cumulative density function , that the future unit selling price is known and independent of the number sold and that non sold units are discarded. We have not considered the possibility of a salvation value , because the only modification is to transform into .

When the actual has occurred, the gain is given by . At ordering time, we have to consider its expected value: . Defining, for a given , the overflow probability , the "lower mean" and the "upper mean" by:

and comparing with the naive value , where , we obtain:

Since everything in this formula is positive, the quantity is an upper bound for and appears to be the cost associated with the choice .

The usual criterion used to determine the optimal value is to minimize this cost. Derivating , we obtain the condition , i.e. the well known:

In other words: when is known, the best quantity you can buy depends on the profitability of the product and is not the expectation of the demand. Reporting Eq. 2 into Eq. 1 leads to the following expression (where is used as index instead of ):

This expression gives the cost of uncertainty, i.e. the cost that remains even when we adopt the best decision.

3.2 Min-max solution for

The best order decision must be analyzed under various weaker hypotheses than an exact knowledge of the distribution . For example, it can be assumed that the mean demand is identifiable with enough precision, and that, additionally, some measure of the dispersion of the demand is also identifiable. In such a condition, various families of demand pdfs need to be investigated. The determination of the optimal decision becomes a min-then-max problem, where the objective is to optimize the gain for the worst case over a family of demand models, in order to guarantee a lower bound for the expected performance. In other words, we solve:

The founding result given in [8] addresses the case where the standard deviation is known (together with the mean ). The key fact is that, over all distributions sharing the 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 distributions, leading to the solution:

It can be noticed that, in the "two Dirac's" case, either or , while the condition on is needed to ensure .

A graphical proof of this result against the family is summarized in Figure 1 (drawn using , , and ). 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 "parabolas" are going through this same point. Therefore, the best decision is the whose curve admits this special point as its minimum.

Fig. 1: Playing against

3.3 Min-max solution for

Knowing that is not the optimal decision, we can nevertheless examine what happens when this choice is taken. By the definition of the mean, we have . Using this expression in Eq. 1, we obtain the "cost of mean" formula:

It can be seen that and are the only two values of for which the "cost of choice" Eq. 1 can be written this way. Obviously, the "cost of mean" is an upper bound for Eq. 3. Being independent of the cost to price ratio, this bound is of interest and places the focus onto the quantity , later referred as the "intermeans parameter" and defined by:

A strange result is as follows. Figure 2 shows what happens when playing against the family of all the "two Dirac's" sharing the same values of the mean and the intermeans parameter. When drawing the curves , the "parabolic" parts are now straight lines but, here again, all of them are going through the same point whose abscissa is Therefore, the robust decision is no more given by Eq. 5 but by:

In other words: if you consider that is the right way to summarize your knowledge 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).

Fig. 2: Playing against