Inventory Management Techniques

How much stock should I keep of a product? Given a delay in ordering time, when should I increase my stock? If I order too much, how do I minimize the amount wasted? These are common, but important questions that lie at the core of inventory management. A discipline that is often mystified with confusing jargon and unrealistic assumptions. In this post, we explore the most common techniques to make inventory decisions in a data-driven way. We will deal with one of the most important variations of the inventory management problem: future demand that is unknown in advance (other variations, such as the case with certain demand, are also important from a logistics and storage point of view and will be covered in another post).

Method 1: Historical Demand Stock Plus a Buffer

The most common and simplest technique for inventory management is to simply look at historical quantities and have in stock the same amount as the last period (year, quarter or week) plus a fixed percentage (2%, 5% or even 10%). So, the inventory estimate $q$ is the historical demand $\mu$ and adding a percentage buffer say, $\delta=5\%$, so that the total multiplier is $1+\delta=105\%$. This inventory rule can be written as

$$ q = \mu (1+\delta)$$

This method is appealing because it’s relatively easy to calculate (you just need to check the historical estimates of previous sales) and the percentage p can be chosen by hand. Not surprisingly, this method can lead to huge losses in certain situations. To illustrate these risks it’s always better to write them numerically:

$$\text{Demand difference} = D - q = D - \mu (1+\delta)$$

The demand difference denotes the amount of lost sales (if $D>q$) or the stock unsold (if $D<q$). It's not difficult to see that both scenarios are undesirable in their own way. If the demand $D$ is greater than the inventory $q$ level, then $D-q$ sales will be lost, if the product has a price p, the lost amount is $(D-q)p$.

On the other hand, when the inventory stock $q$ is greater than the demand $D$, the amount of products unsold is $q-D$. Depending on the type of product, there will also be some amount of money lost per product, either because of expiration, storage, liquidation or other costs. Say that on average this cost is $k$ dollars per product, then $(q-D)k$ dollars will be lost.

With this framework it's easy to see the tradeoff and nuance of the inventory decision. If the future demand is unknown, then it's very hard only with this information to know what's the optimal buffer multiplier \(\delta\), because in the case when the historical demand $\mu$ under or over estimates the actual demand, the optimal multiplier would be:

$$\delta = \frac{D}{\mu}$$

But clearly the buffer quantity $\delta$ is impossible to estimate without knowing the random future demand $D$. So, even though this seems like a reasonable method, it is in fact the same as randomly guessing the future demand.

Pros:

• Easy to estimate (only requires a guess for the buffer and the historical demand)

Cons:

• It is the same as blindly guessing the demand, potentially leading to either losses by unsold products or being short of the actual demand.

Method 2: Statistical Optimization Method (The Newsvendor Model)

The previous method, while reasonable and intuitive, seems to have a missing ingredient. The missing ingredient lies in somehow quantifying and balancing the two asymmetric scenarios mentioned in the previous method (of either excess inventory or being short of the unknown future demand). To solve this seemingly impossible problem, we use probability theory. For those unfamiliar with probability, it is simply a way to model mathematically the randomness of the demand $D$ (emphasis in modeling, as there is no way to know the future!).

In probability theory, the demand $D$ is considered a random variable. To simplify the analysis, we suppose the demand is normally distributed the mean $\mu$ and standard deviation $\sigma$. In broad terms, this means that in average the demand $Q$ is around the mean $\mu$ and that the average deviation from the mean is $\sigma$. Moreover, the randomness is concentrated around the mean, so either too large or too small outcomes are unlikely. You will see next why it is useful to model the demand this way.

Going back to the previous discussion, the profit can be written as:

$$\text{profit} = \min (D,q)p - qp$$

Which is to say that the smaller between the demand $D$ and the inventory $q$ will be sold at a price $p$, that will be the revenue, and each product has a cost $p$, therefore, a production/purchase cost $qp$. Omitting the math behind solving the problem (which is interesting in its own), leads to the following inventory formula:

$$q = \mu + \sigma\phi^{-1}\left(\frac{p-c}{p}\right)$$

This rule means that the optimal inventory quantity is the historical average $\mu$ plus a buffer which is the product of the standard deviation of the demand $\sigma$ multiplied by the quantile function of a standard normal evaluated at $\frac{p-c}{p}$ (in principle all of these functions can be estimated in excel, $\mu$ can be computed with AVERAGE, $\sigma$ with STDEV.S and the quantile function $\phi^{-1}$ with NORM.S.INV). This method is connected to the previous one in the sense that this one gives an explicit amount for the demand buffer that is directly connected to the variation of the demand $\sigma$ and also to the normalized profit margin per sale $\frac{p-c}{p}$. The higher the margin, the more painful it is to lose a sale and therefore the more inventory is needed. There are many variations of this formulation (it is called the Newsvendor model in an influential paper in Econometrica by Kenneth Arrow, T. Harris, and Jacob Marshak and has its origins in an 1888 paper by Edgeworth to find the optimal bank cash reserves) where other scenarios can be modeled, such as the one where the overage incurs another cost.

Pros:

• Still easy to estimate once all the input quantities are given.
• Modeling flexibility to other scenarios

Cons:

• While on a surface level it looks like the problem is solved, estimating the average future demand $\mu$ and its standard deviation $\sigma$ can be hard when there are seasonal components and other factors that affect the demand. We cover some of these in this post. In general, many instances of this problem need to be solved continuously to effectively manage the inventory.

Conclusion

In this post we explored two methods that are closely related for estimating optimal inventories. Both rely on the idea of calculating and extra buffer to account for uncertainty on the demand. Like everything, the devil is in the details of estimating the future demand either by hunch or using statistical methods. At Predictheus, we have a platform that calculates these automatically from many sources of data using state-of-the-art statistical and computational methods in a seamless way.