Can you think of a product more perishable than the traditional newspaper? Today’s newspaper cannot be sold tomorrow! No one wants to buy yesterday’s newspaper. The newsvendor has to discard his old news papers or sell them as scrap. 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
- News Vendor Problem Example (without Salvage Value)
- The Intuition Behind the Newsvendor Model: Too Much vs. Too Little or Critical Ratio
- The Critical Fractile or Critical Ratio
- Applying The Critical Fractile to Demand with Different Distribution Patterns
- Testing the News Vendor Model with a Simulation
- Newsvendor Model Performance Measures
- News Vendor Problem Example (with Salvage Value)
- Newsvendor Model Practice Questions
- Profit Maximizing vs. Target Service Level Approach
- Operations Research Tutoring for Newsvendor Models
Applications of the Newsvendor Model
The setting used to formulate the news vendor model may have been a humble news vendor’s challenge, but many businesses face this problem. Most MBA students will encounter the newsvendor problem in their first operations management class, given it’s possible application in a wide range of industries. Some of the situations where the newsvendor model can be applied include:
- How many 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.
The 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. If you know what the demand is, you do not have to worry about how many units to order. When demand is uncertian, the newsvendor model help you decide how many units to produce taking into consideration the cost of having too much and the cost of having too little.
The 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 decide on production or purchase only once in a season or period. In other words, if you can reorder to fulfill the season’s demand when you have no inventory left, then you do not have to worry about the quantity to order at the beginning of the season. The newsvendor model is very valuable when you have to make the stock or inventory decision ahead of the season and when you cannot reorder causing you lost sales for the season if demand exceeds the quantity ordered.
News Vendor Problem Example (without Salvage Value)
Let’s say that the news vendor sells about 100 papers a day. Unfortunately, demand is uncertain. On some days he sells as many as 150 papers and other days he sells as few as 80 papers. On analyzing this news vendor’s demand, let us assume that we found that the demand is normally distributed with a mean of 100 newspapers and a standard deviation of 20 newspapers. 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 of how much to order as an example.
The Intuition Behind the Newsvendor Model: Too Much vs. Too Little
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 News Vendor Model Tradeoffs: Too Much vs. Too Little
The approach to solving the news vendor problem is balancing the cost of ordering too much vs. the cost of ordering too little.
The “Cost of Overage” Or Co
The news vendor model 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. Cost of overage is also referred to as cost of excess inventory, cost of stale stock, etc. 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 news vendor model 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. Note that cost of underage is also referred to as cost of lost sales, cost of understocking, cost of unfullfilled demand, etc.
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, loss of repeat customers, cost of rush orders, backorder possibilities, must be considered when computing the cost of underage or Cu in more complicated newsvendor model scenarios.
The Critical Fractile or Critical Ratio
So the critical fractile tries to balance the cost of overage, Co, and the cost of underage, 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 the 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 Demand with Different Distribution Patterns
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.
Applying The Critical Fractile to Demand with Normal Distributions
In our newsvendor problem example, we assumed that the demand is normally distributed with a mean of 100 newspapers and a standard deviation of 20 newspapers. We also found that the critical fractile or critical ratio using the formula Cu/(Co+Cu) works out to be 0.8 (=0.8/(0.8+0.2). This means that the newsvendor model recommends keeping stock so that we can meet demand with a probability of 80%.
We can arrive at the recommended stock level when the distribution is normal, using the formula
Recommended Stock Level = Mean of Demand + Z(80%) * Standard Deviation of Demand
The z value for an 80% level can be arrived at using the =norm.s.inv(80%) excel formula or looking up the standard normal distribution table. Either way, you will arrive at a z-value of 0.8416.
Substitutinr the mean, standard deviation and z values in the recommended stock level formula we get
Recommended Stock Level = 100 + 0.8416 * 20 = 116.832 newspapers.
Or we can also use the norm.inv(probability, mean, standard deviation) formula to arrive at the above answer.
Applying The Critical Fractile to Demand with Uniform Distributions
The above formula was applicable if the demand for newspapers was normally distributed. What if the demand was uniformly distributed? Our approach will have to be different. Let us learn how our approach will be different using an example.
In our newsvendor problem example, let us assume that the demand for newspapers can vary anywhere between 10 and 80 newspapers uniformly. Here, if we assume the lower limit of a=10 newspapers and an upper limit of b=80 newspapers, the formula is:
Recommended Stock Level = a + critical fractile * (b-a)
Filling in the numbers for the newsvendor model sample problem without salvage value, recommended stock level can be arrived at as follows:
Recommended Stock Level = 10 + 80%* (80-10) = 66 newspapers.
Assuming a uniform distribution, we know that we must stock up to 66 newspapers (=10+.8*70) according to the newsvendor problem model.
Applying The Critical Fractile to Demand with a Custom Distribution
If the distribution is a custom distribution, we will apply the critical fractile by targeting a probability of meeting the demand up to the critical fractile. Let us apply this to our newsvendor problem example without salvage value if the distribution of demand for newspapers is as follows (custom distribution).
Since we found a critical fractile or critical ratio using the formula Cu/(Co+Cu) of 0.8 (=0.8/(0.8+0.2) we will target a level of inventory such that we will meet demand with a probability of 80%. To find the level of stock that will meet demand with a probability of 80% we will build the frequency table below from the custom distribution of demand for newspapers to arrive at the cumulative percentages.
We will stock to a level that satisfies demand up to a probability of meeting the demand as close to the critical fractile of 80%. In our newsvendor problem example without salvage value, we will stock 140 newspapers and meet demand with a probability of 79.3%.
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 with illustrations so they get sufficient practice applying the model.
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 simulation software such as @Risk or Crystal Ball or languages such as Python and R. 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 or programming languages such as Python and R.
Newsvendor Model Performance Measures
The new vendor model is applied in many HBS case studies encountered in an operations research class in MBA programs. You can also evaluate a company’s target inventory stocking levels 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
News Vendor Problem Example (with Salvage Value)
Earlier, we looked at a newsvendor problem without salvage value. Now, let’s look at a newsvendor problem WITH salvage value. Let’s also that the newsvendor buys a paper for $0.20 and sells it for $1. Unsold stock can be sold for $0.10. 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. Let’s say that the news vendor sells about 100 papers a day. But demand is uncertain and is normally distributed with a mean of 100 newspapers and a standard deviation of 20 newspapers. The news vendor must make his inventory decision early in the morning before he knows how many newspapers he will sell. How will you help the news vendor make this decision?
Let us use the following notations and assumptions for the newsvendor example with salvage value:
- c = $0.20 = cost per unit ($)
- p = $1.00 = price per unit or revenue per unit ($)
- s = $0.10 = salvage value per unit ($)
Then the cost of underage or Cu and the cost of overage or Co can be worked out as follows:
- Cu = (p-c) = $0.80 Cost of underage
- Co = (c-s) = $0.10 Cost of overage
Therefore the critical fractile works out to be 0.89 (=0.8/(0.8+0.1).
Given that demand is normally distributed with a mean of 100 newspapers and a standard deviation of 20 newspapers, we can find the target inventory level that maximizes profits using the formula:
Recommended Stock Level = Mean of Demand + Z(89%) * Standard Deviation of Demand
Or we can use the NormInv Microsoft Excel function.
Recommended Stock Level = norm.inv(probability, mean, standard deviation)
Either way we will arrive at a target inventory level of 124 .41 newspapers to maximize our profits.
Newsvendor Model Practice Questions
Here are some more news vendor model questions for practice to test out your understanding of the news vendor model:
- A drone firm buys its motors at $75 each from an overseas vendor. So orders have to be placed only once a season. These motors are sold at a retail price of $160. If unsold, these motors have to be used in older models and sell for $20 per unit. Assuming a demand is normally distributed with a mean of 800 and a standard deviation of 400, how many should the firm order for this season? What is the expected profit? What is the probability of not meeting demand?
- Noodles is sold for $2 per pack. It costs $1/pack. Unsold units are discounted to $.4/unit. Demand has a mean of 400 units with a standard deviation of 120 units. How many packs should you order? What is expected sales? What is the expected lost demand? What is expected profit?
- TRY is a toy manufacturer. Try has to decide on the quantity to make in the last production run for this season. Any fewer than the customer demand causes it to lose the profit of $28 from the lost sale and possible loss of goodwill valued at $6. If too many are made, the cost of manufacturing of $16 is lost. How many units should they make if demand is expected to be 500 units with a probability of 10%, 600 units with a probability of 15%, 700 units with a probability of 15%, 800 units with a probability of 30%, 900 units with a probability of 20%, and 1000 units with a probability of 10%?
- A new hat is sold for $29 per hat. It costs $20/hat. Unsold units are discounted to $12/hat. Demand is expected to be between 6000 and 15000 hats. How many hats should Brown order? What is expected sales? What is the expected lost demand? What is expected profit?
- Verna sells Bytees imported from Peru only once a year. She makes $45 in profit on the sale of a new Bytee. Whereas any unmet demand costs Vema $30 including lost goodwill. How many Bytees should Verna stock before Thanksgiving for the next Christmas season if demand is expected to be 5,000 units with a standard deviation of 350 units?
Profit Maximizing vs. Target Service Level Approach
The news vendor model approach is maximizing profits by balancing the cost of having too much vs. the cost of having too little. Often, the objective of the organization may be achieving a target service level. The target service level approach is setting the goal of having sufficient stock a specific percentage of the time. For example, a business may say we want to have sufficient stock available 95% of the time or 99% of the time.
The newsvendor model maximizes profits and does not consider the target service level level. For example, in the sample news vendor problem without salvage value, we found the critical ratio of 80%. This means we would only stock up to a level where we meet demand 80% of the time, or the probability of meeting demand on any given season or period is only 80%. In other words, we will not meet demand 20% of the time. Management may decide to override the newsvendor recommendation and stock up at higher levels to meet demand 95% or 98% of the time, but must be aware that the profit-maximizing inventory levels have been breached.
Note that if the cost of lost customers or cost of stockouts is correctly determined, then the newsvendor model can also help you determine the right target service level because it is trading off the cost of having too much and the cost of lost customers!
Operations Research Tutoring for Newsvendor Models, Etc.
Our operations research tutors can assist you with tutoring for the newsvendor model and other inventory management case studies. Other operations topics we can assist you with include queuing theory and waiting lines, decision trees, linear programming using Microsoft Excel’s Solver, newsvendor models, batch processing, Littlefield simulation games, etc. Feel free to call or email if we can be of assistance with live one-on-one tutoring.