This post introduces the Newsvendor inventory management model and walks through usage, decision criteria, inputs, and tradeoffs. We also dive into manufacturing luxury nunchucks because we all know that's an untapped gold mine. Upcoming blog posts will explore EOQ and Q/R inventory management models as well.

For additional resources, check out:

- The Ondema Guide to Production Scheduling
- Minimizing Makespan, Explained Using Attack Helicopters
- Level Up Production Bottleneck Hunting With These Tips
- Manage Production Queues Like a Boss
- Inventory Management Models Explained With Nunchucks - Pt. II
**Inventory Management Models Explained With Nunchucks - Pt. III**

## Why is managing inventory a big deal?

Inventory affects almost every aspect of your business's financial position. Business revenue is driven by product availability being closely matched to demand. Just-In-Time or Continuous Flow distribution frameworks can be the largest drivers of Cost of Goods Sold (COGS) and expenses.

In short: good inventory management is critical to a business's survival, let alone profitability.

In this post, we'll touch on the first of three inventory management models: 1. Newsvendor, 2. QR, and 3. EOQ. Not only are they industry-proven, but they form the foundation for almost every inventory management software solution on the market, even the ones with seven-figure price tags.

## The Newsvendor model

Decisions around inventory models boil down to three main questions:

- How often should inventory status be monitored?
- When should replacement inventory be ordered?
- What quantity of replacement inventory is optimal?

The Newsvendor Model helps you chose *Q*, the stocking quantity. The caveat is that it is chosen before demand is known with the goal of maximizing expected profit. The only thing we all know about forecasts is they're wrong; this model does a nice job addressing demand uncertainty.

The name comes from (you guessed it!) newsvendors choosing the quantity of periodicals to stock at a news stand. In practice, it's used for innovative products (e.g., a new fashion line) where demand is uncertain over a period of time and inventory can't be replenished during the selling period.

Demand is uncertain but is assumed to be normally distributed. What does this mean? It helps us understand the probabilities of certain demand scenarios. For example, stocking the "average" number of newspapers for any period means that you would be 50% sure that you'd "have enough" to satisfy demand.

In the graph below, *μ* is the average and *σ* is standard deviation.

Credit: Galarnyk (2018).

Let's say demand over a one day period averages 30 newspapers (*μ*) with a standard deviation of 5 (*σ*).

- If you choose to stock 30 newspapers for the coming day, you have a 50% probability of having enough inventory to satisfy demand.
- If you decide to stock 35 newspapers (
*μ*+*σ*), you have a 50% + 34.13% = 84.13% probability of having enough inventory to satisfy demand. - If you decide to stock 40 newspapers (
*μ*+ 2*σ*), you have a 50% + 34.13% + 13.59% = 97.72% probability of having enough inventory to satisfy demand. - Etc.

The newspapers above the daily average are referred to as "safety stock." Our total stocking quantity can be mathematically expressed as average demand plus safety stock:

Q = *μ* + *σ* * Z

With Z being the number of standard deviations above the average demand during replenishment lead time required for a desired service level.

## So what should the service level be?

Before we dive into that, it's important to note there are two different definitions for service level:

*Type 1 (also known as α service level)*- measures the probability that*all*demand arriving within a given time interval will be completely delivered from stock on hand. In other words, it's the probability of no stockout.*Type 2 (also known as β service level)*- the proportion of total demand within a reference period which is delivered without delay from stock on hand. This is also known as the "fill rate."

So how can we go about setting Type 1 service levels? By comparing the cost of overstocking (C_{o}) to the cost of understocking (C_{u}), of course!

C_{o} is the cost of ordering one more unit than was needed to completely satisfy demand. It's the increase in profit that would have been realized had one *less* unit been ordered. Conversely, C_{u} is the cost of ordering one less unit that was needed to completely satisfy demand. It the increase in profit that would have been realized had one *more* unit been ordered.

In general, ordering one more unit increases the chances of overstock, but reduces the chances of stockout. Also, as more units are ordered, the expected gain from ordering one more unit goes down, while the expected loss of ordering one more unit goes up.

##### BEHOLD, the Critical Ratio!

What does this equation tell us? To maximize profit, choose *Q* such that we don’t have lost sales (i.e., demand is *Q* or lower) with a probability that equals the critical ratio.

## Dan's Luxury Nunchucks™

Let's walk through an example. Let's say I make luxury nunchucks (LuxChucks™) because I'm an innovator leaning into the rich-people-pay-a-lot-for-ninja-weapons market. Who *doesn't* want to look bad-ass while practicing ninjitsu in their Crocs with the garage door open blasting "Eye of the Tiger" on repeat??!?! NOBODY, that's who.

Sometimes my genius is almost frightening. But I digress.

Raw material and labor costs to manufacture these one-of-a-kind nunchucks total $300. I sell them for $1,000. Average demand is 40 units per month with an estimated standard deviation of 7.

What's the correct service level and quantity to stock? C_{u} is the gross margin I'd miss out on - $700. C_{o} is the loss I'd incur if one unit isn't sold - $300.

In order to achieve a service level of 0.7, I should stock 44 luxury nunchucks.

##### A quick recap of the Newsvendor model, so far

- It's good for scenarios where a single production or replenishment order is appropriate, demand is unknown, and a "too-much vs. too-little" challenge exists.
- Manufacturers should use a demand model that quantifies expected demand AND uncertainty of that demand.
- The expected profit-maximizing amount
*Q*balances the tradeoffs between overages and underages.

## Newsvendor model performance measures

Of course things always change, so we need metrics to measure the performance of our Newsvendor Model.

##### In-stock probability

This is the probability that all demand is met.

##### Expected lost sales

This is the average number of units if demand exceeds the order quantity. More specifically, it is the average over all possible demand outcomes. It can be represented as:

Let's walk through an example. Let's say annual demand for Dan's Luxury Nunchucks™ is normally distributed with a 1,100 average and standard deviation of 300. I've chosen to stock 1,200 luxury nunchucks.

Our first step is to normalize the order quantity to find its z-statistic. Then we use Excel to find the expected lost sales for a standard normal distribution with that z-statistic. Then we evaluate lost sales for the actual normal distribution. All of this looks like:

Expected lost sales is 51 nunchucks.

##### Expected sales

This is the average number of units sold. Expected sales is simply the average *μ* minus expected lost sales. In this example, that number is 1,100 - 51 = 1,049 nunchucks.

##### Expected left-over inventory

The average number of units left over at the end of the selling period. It's the chosen stock quantity *Q* minus expected sales. In this example, that number is 1,200 - 1,049 = 151 nunchucks.

##### Expected fill rate

This is the percentage of demand that is taken care of immediately. This is calculated as:

Some of you may be saying this right now.

While you may be praying for a merciful end to this post, there's a few more things worth touching on.

## Reducing uncertainty

Reducing uncertainty helps businesses achieve more accurate stocking quantities. In other words, tightening up the variance of demand forecasts increases profits.

So what are techniques for reducing the variance of demand and/or demand forecasts? DeHoratius (2015) summarizes it nicely:

- Learn as much about demand as possible before committing to production
- Share demand data throughout your supply (value) chain
- Vendor-managed inventory
- Reduce the forecasting horizon
- Avoid unnecessary trade promotions that lead to retail-forward buying that disguises true consumer demand

## That's all for today, folks

We covered a lot of ground in this post. This week we'll also detail the EOQ and Q/R inventory management models. *Hooray!*

Until next time,

DB

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