### Understanding Newsvendor model through an example

Newsvendor formula is pretty powerful formula, if used properly. Before we delve deeper into the best ways to leverage the new vendor model, it makes sense to understand the basics of newsvendor model.

Say I decide to publish my blog daily in print format, in the form of a newsletter and price the newsletter $1. Now I don’t know exactly how many newsletters I will be able to sell in a day. But I do need to decide how many I want to print. Assuming that I am selling these newsletters with the intention of making money, I want to maximize my revenue and minimize my losses.

Now let us assume that it costs me $0.1 to print one newsletter. So if I sell one (assuming there are no other costs involved), I make $0.9 ($1-$0.1 printing cost). If there are unsold copies though, I am facing loss at $0.1 (printing cost) for each copy. So I do want to sell all I print to maximize my revenue.

**Printing too much or too less**

So I look at my sales figures for last six months and see that the minimum that I have sold in last six months is 10/day. So if I print only 10 copies, the chances of me being left with unsold copies is minimal. However, what about the lost sales? I find out after few days that at least 5 individuals are coming to buy the newsletter after I have sold my ten copies. So essentially, if I had five more copies, I could have easily made $0.9 X 5 more. So printing only 10 copies is not maximizing my revenues.

Essentially, what I have is a problem of “Too much or too little”? So the question then is how do I figure out what is the “optimal” quantity I should print each day, given the following data points that I have:

- Historical sales data
- Cost of losing
**a**sale ($0.9) - Cost of not selling/copy ($0.1)

Given the dilemma, I want to use a scientific methodology to determine to determine what is the quantity that I should print that will help me maximize my profits.

**The Newsvendor formula**

Below is the newsvendor formula. Now let us go through the components of the formula to understand it better.

As stated in my example above, the formula can help me calculate what is the optimal quantity of newsletters that I should print that is “optimal”. The notation F(Q) represents the probability that the demand is less than or equal to Q. So the Q that you determine using this probability value is the optimal Q. This will become more clear once we go through an example.

Now let us go through each of the parameters indicated in the formula:

** = Cost of underage.**

So if we go back to my newsletter example, if after I sell all my newsletter, there is still demand for my newsletters, for each unit of demand, I am losing $0.9 ($1 selling price – $0.01 printing cost). This $0.9 is my cost of underage, revenue lost due to under producing

=** Cost of overage.**

Going back to the example again, If there are unsold copies though, I am facing loss at $0.1 (printing cost) for each copy. This cost is the cost of overproducing, hence referred to as cost of overage.

The value of F(Q) thus calculated using the formula above is: 0.9

The value of F(Q) calculated above is basically providing you your ** optimal in stock rate or the optimal service level, e**xplained in more detail in a subsequent paragraph.

**Calculating my optimal print copies**

So now, we know that the optimal quantity will help me provide a service level of 90% but we still don’t know what that quantity should be.

This brings us to the topic of * demand distribution*. This is a very intensive topic in itself …there are links at the end of this article to resources you can use to gain fundamentals on this topic. But in very simple terms, if I plot my historical demand , it looks something like the chart below:

**Optimal Service Level**

**But what does this optimal service level number mean?**

So what does that 0.9 mean ? The reason this parameter is denoted by F(**Q**) is because F(Q) is the probability associated with an optimal stocking (in this case printing) quantity Q (and we will discuss how we can calculate that Q from the F(Q) we calculated in a later section). So we can decipher 0.9 as:

*If I print Q newsletters everyday, my probability of fulfilling all the demand is 90%. Q is the optimal number of newsletters I should print and the corresponding service level that I will provide with that quantity is 90% *

**The z factor**

Where z is the number of standard deviations you need to add to average demand to obtain your desired in-stock rate F(Q).

Each service level that we have calculated above corresponds to a value of the z factor (number of standard deviations) and can be determined using a reference table that maps z value and service level/probability *for each distribution type.*

**Modeling historical demand accurately is **critical

Since the value of z, derived from the value of F(Q), is dependent on the type of distribution, the key to doing this accurately is your historical demand data.

Calculating an accurate empirical distribution is very critical for optimal application of newsvendor model since the properties of that distribution, along with the probability calculated above, will help us determine the optimal number of newsletters I should print.

In my case, I have determined that my newsletter demand is indeed a normal distribution with daily Mean of 10 newsletters and standard deviation of 2 newsletters.

Now, I used the z value and probability table for normal distribution to get z value for a probability of 0.9

z = 1.282

**Finally, calculating the quantity**

The demand distribution that I created in the prior section is a specific type of distribution (Normal Distribution). As mentioned above, each distribution can be defined by certain parameters. For Normal Distribution, the Quantity for a specific z value is defined by the following formula:

Now we can use the above formula to determine the optimal newsletter quantities that I can print. Substituting the values in the formula above:

Q = 10 + 1.3 X 2 ~13 Newsletters daily