Newsvendor Model

The newsvendor model (or the single-period model) is named, simply, after how a newsvendor would have to stock up. That is, in the morning, they have to decide how many newspapers they need to have on hand to meet demand that day. While they are on their run, they can’t go back to the warehouse. As such, the newsvendor model is a game of trying to predict what your demand will be, and making sure in the majority of cases (e.g. 99% of the time), you have enough inventory to meet demand, while minimizing the amount of excess inventory you have on hand.

Background

Using the model, we assume that demand follows a normal distribution. As such, let’s say we have a mean demand of 100 units, and a standard deviation of 20 units. If we choose to stock 120 units, then we can calculate our in-stock rate as the probability that the demand is less than or equal to 120 units, which is 84.13%. If we want to increase our in-stock rate to 95%, then we need to stock 132 units.

Using the Model

But, alas, such is not quite as simple. In practice, there are two competing costs that we are trying to balance:

• The cost of having one less unit than customers demand (i.e. the per-unit cost of underage)
• The cost of having one more unit than customers demand (i.e. the per-unit cost of overage)

For our specific model, we calculate those two numbers as follows:

• cost of underage $c_u=\text{lost gross margin per unit}=\text{selling price per unit}-\text{cost per unit}$
• cost per overage $c_o=\text{cost per unit}-\text{salvage value per unit}$

Note: we ignore more difficult-to-quantify costs such as customer loyalty that might result from stockouts. This, you should talk about qualitatively

Now, it’s a game of optimizing to minimize both of these costs. That is, we want to find the perfect amount to stock such that we minimize the expected cost of underage and overage. After some math, we find that the optimal in-stock rate is given by the following formula:

$$\text{Optimal In-Stock Rate}=P(D\le Q^{\ast })=\frac{c_u}{c_u+c_o}$$

The woke gang will find this eerily similar to calculating pot odds in poker.

Then, to calculate the actual value of $Q^{\ast }$, we need to use NORM.INV in excel.

Q* = NORM.INV(Optimal In-Stock Rate, Mean Demand, Standard Deviation of Demand)