# Newsvendor Model Problem & Example

Can you think of a product more perishable than the traditional newspaper? By the next day, the news is pretty much worthless! No one wants to buy yesterday’s newspaper. The newsvendor has to discard his old paper or sell it as scrap paper. This is a big problem for the newsvendor!

The news vendor’s problem is that he must decide how many newspapers to buy at the beginning of the day. He must make his decision before he knows how many newspapers he will sell. And thus was born the newsvendor problem leading to the newsvendor model.

## Newsvendor Model Applications

The setting used to formulate the newsvendor model may have been a humble news vendors’ challenge but this problem is faced by many businesses. Most MBA students will encounter the newsvendor problem given it’s possible application in a wide range of industries. Some of the situations where the newsvendor model can be applied include:

• How much fresh vegetables should I take to the weekly farmers market today?
• How much bread should I bake for today’s sales?
• How many Christmas trees should I cut for this season?
• How many units should we make of this season’s summer dresses?

The newsvendor model is used to address these questions. The newsvendor model applies itself nicely to the following general situations.

### Newsvendor Model is Applicable when Demand is Uncertain.

The newsvendor model helps you decide how many units to produce or buy when demand is uncertain taking into consideration the cost of having too much and the cost of having too little.

### Newsvendor Model is Applicable for One-shot or Seasonal Decisions

The newsvendor model also helps you decide how many units to produce or buy when you can you have to decide on production or purchase only demand is uncertain taking into consideration the cost of having too much and the cost of having too little.

## News Vendor Problem Example

Let’s say that the newsvendor sells about 50 papers a day. Unfortunately, demand is uncertain. On some days he sells as many as 80 papers and other days he sells as few as 10 papers. Let’s also assume that the newsvendor buys a paper for \$0.20 and sells it for \$1. The news vendor’s problem in this example is that he must decide how many newspapers to buy paying \$0.2 per copy at the beginning of the day. He must make his decision before he knows how many newspapers he will sell. We will show you how to solve this News Vendor’s Problem as an example.

## The Intuition Behind the Critical Fractile

The problem facing the newsvendor is two dimensional: On one side, if he orders too many newspapers, he will be left with unsold newspapers which reduces his profits. On the other hand, if he orders too few newspapers, he is losing out on the profit he could have had if he had newspapers to sell. The newsvendor model tries to balance these two competing forces of having too much and having too little. The newsvendor model balances these two forces using the critical fractile.

### The “Cost of Overage” Or Co

The critical fractile defines the cost of having too much as the “cost of overage” or Co. Overage here indicates stocking more units than the demand of the relevant period. In our example, the newsvendor buys a newspaper for \$0.20. So if he has newspapers remaining unsold, he loses \$0.20 for every unsold newspaper. This is his cost of overage.

Other aspects such as salvage value, inventory holding costs, the opportunity cost of capital, etc must be considered when computing the cost of overage or Co in more complicated newsvendor model scenarios.

### The “Cost of Underage” Or Cu

The critical fractile defines the cost of having too little as the “cost of underage” or Cu. Underage here indicates not having enough units to meet the demand of the relevant period. In our example, the newsvendor buys a newspaper for \$0.20 and sells it for \$1. So if he does not have newspapers to sell when a customer requests one, he loses \$0.80 of profits (\$1 revenue less \$0.2 of cost) for every unit of unmet demand. This is his cost of underage or Cu.

Other aspects such as lost goodwill and repeat customers, cost of rush orders, backorder possibilities, etc. must be considered when computing the cost of underage or Cu in more complicated newsvendor model scenarios.

## The Critical Fractile

So the critical fractile tries to balance Co and Cu. the critical fractile does this using the critical fractile formula below.

The critical fractile = Cu/(Co+Cu)

The crux of the newsvendor model is the critical fractile. In our newsvendor model example, we worked out the cost of underage or Cu of \$0.80 and a cost of overage or Co of \$0.20 above. Therefore the critical fractile works out to be 0.8 (=0.8/(0.8+0.2).

## Applying The Critical Fractile to Different Distributions

Once the critical fractile is arrived at, inventory is ordered up to that point on the demand distribution. This is the level of inventory that meets the demand for the critical fractile percentage of times. This level of inventory is the point at which we maximize demand.

In our newsvendor problem example, we assumed that the demand can vary anywhere between 10 and 80 newspapers. Assuming a uniform distribution, we know that we must stock up to 66 newspapers (=10+.8*70) according to the newsvendor problem model. If the distribution was normal or a custom distribution, we will apply the critical fractice computed in our newsvendor problem example accordingly resulting in a probability of meeting the demand up to the critical factor.

As graduate-level tutors, we encounter students trying to understand the intuition behind the critical fractile often. Our tutors walk them through the construction of the critical factors using numbers so students pick up the intuition behind the critical fractile computation. Our MBA tutors also show students how to use the critical fractile on a variety of demand distributions.

## Critical fractal vs. Critical Fractile

The word fractile is defined in statistics as the value of a distribution for which some fraction of the sample lies below and therefore is the better word to pick up the concept of critical fractile. We have seen faculty use the word fractal. The definition of fractal is a curve or geometric figure, each part of which has the same statistical character as the whole. While this may be OK, we believe that the critical fractile is a better term to use.

## Testing the News Vendor Model with a Simulation

Often students do not really believe that the newsvendor model works. They find it difficult to believe the newsvendor model because it uses the critical fractile formula which is so simple. We help address this doubt by simulating the newsvendor model using @Risk or Crystal Ball. We use the risk optimizer or data table functions to show the maximum profit appearing at the inventory levels indicated by the critical fractile.

### News Vendor Model and Industry/Supply Chain Dynamics

We also show students how the newsvendor model impacts policy making, decision making, and profitability in an industry. This may occur as part of a case study. If it is not covered in a case study, we show how a supplier and buyer can work together to form policies that enable more profits for both parties. We also show this using simulation using @Risk or Crystal Ball.

### Newsvendor Model Performance Measures

Often the new vendor model is applied in HBS case studies. You can also evaluate the Newsvendor model using the following metrics.

• Expected sales = average number of units sold.
• Expected value of lost sales = average number of demand units that exceed the order quantity
• Fill rate = usually the critical fractile which is the fraction of demand that is met (sometimes referred to as in-stock probability or the probability all demand is satisfied
• Stock out probability = Probability that demand exceeds the inventory and is not met.
• Expected unsold inventory = the average number of inventory units that exceed the demand Expected profit = Expected sales – corresponding costs

Please do reach out if we can assist you with tutoring for the newsvendor problem.