The demand for next year is forecasted to have a mean of 10,000 units

The demand for next year is forecasted to have a mean of 10,000 units (with a normal distribution), and its standard deviation is 2,000. Let us assume that if the demand is more than what we produce and we are out of stock of this item, we lose $30 per piece; however, if we demand less than what we produce, each item left unsold costs us $20 (we have to put this on sale). How much should we order from our offshore supplier under these circumstances?

We should order an amount such that the expected cost of overstocking is equal to the expected cost of under stocking.

The normal distribution, shown below, represents the demand distribution. The average demand (corresponding to the 0 in the diagram) represents 10,000 units. Each unit on the horizontal axis represents one standard deviation of the distribution. In this case, each unit represents 2,000 units.

Let us assume that we order the production of an amount represented by the red line, which is z units to the right of the mean.

The area under the normal curve to the right of the line represents the probability (p) that the demand will be more than the production, whereas the area to the left represents the probability (1 – p) that the demand will be less than the production. (Recall that the total area under the normal curve is 1.0.)

Probability that a unit will not be available for sale is p, while the probability that a unit will remain unsold is 1 – p.

We want the expected cost of overstocking to be equal to the expected cost of under stocking.

That is,      p * 30 = (1 – p) * 20

Solving for p results in p = 0.4. The area to the left of the line is, therefore, 0.6.

Referring to standard normal tables, we obtain z = 0.2257.

The desired production, therefore, is 10,450 (10,000 + 2000 * 0.2257).

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