# Accuracy gains (for low turnover) and financial impact By Joannes Vermorel, February 2012

Update (2019): the perspective presented in this article is dated to some extend. This article adopts a classic forecasting perspective, while probabilistic forecasting should be considered instead, as they yield better result in nearly all supply chain situations. In particular, the economic perspective on the forecasting accuracy is better tackled through approaches like the stock reward function.

More accurate demand forecasts generate savings as far inventory is concerned. This article quantifies savings for inventories with turnovers lower than 15. We adopt a viewpoint where the extra accuracy is entirely invested in lowering inventory levels while keeping stockout rates unchanged.

For inventories with higher turnover, we suggest to use our alternative savings formula where the extra accuracy is invested in lowering stockout rates while keeping the inventory levels unchanged.

## The formula

The detail of the proof is given below, but let's start with the final result. Let's introduce the following variables:
• $V$ the total inventory value.
• $H$ the yearly carrying cost (percentage), which represents the sum of all the frictions associated to the inventory.
• $\sigma$ the forecast error of the system in place expressed in unit MAE (mean absolute error). The definition of this measure is given below.
• $\sigma_n$ the forecast error of the new system being benchmarked (hopefully lower than $\sigma$).

The yearly benefit $B$ of revising the forecasts is given by: $$B=V H \left(\sigma - \sigma_n \right)$$

## Unit MAE

The formula introduced here works as long as errors are measured over the lead time and made homegeneous to a percentage with respect of the total sales during the lead time.

Although the MAPE (Mean Absolute Percentage Error) measured over the lead time would fit this definition, we strongly advise not to use the MAPE here. Indeed, the MAPE gives erratic measurements when slow mover's are present in the inventory. Since this article focuses on inventories with low turnover, the existence of slow mover's is a quasi-certainty.

In order to compute the unit MAE (i.e. homogeneous to a percentage), let's introduce:
• $y_i$ the actual demand for the item $i$, for the lead time duration.
• $\hat{y}_i$ the demand forecast for the item $i$, for the lead time duration.

For the consistency of the measurement, we assume that the same starting date $t$ is used for all items. Then, for a set of items $i$, the unit MAE could be written as: $$\sigma = \frac{\sum_i |y_i - \hat{y}_i|}{\sum_i y_i}$$ This value is homegeneous to a percentage and behaves essentially like the MAE. Contrary to the MAPE, it is not negatively impacted by slow mover's, i.e. items where $y_i = 0$ for the period being considered.

## Practical example

Let's consider a large B2B retail network of professional equiments that can obtain a 20% reduction of the relative forecast error through a new forecasting system.

• $V = 100,000,000$ € (100 millions Euros)
• $H = 0.2$ (20% yearly friction cost on inventory)
• $\sigma=0.2$ (old system has 20% error)
• $\sigma_n=0.16$ (new system has 16% error)

Based on the formula above, we obtain a gain at $B=800,000$€ per year.

## Proof of the formula

In order to prove the result given here above, let's introduce introduce a systematic lowering bias of $\sigma - \sigma_n$ percents to all forecasts produced by the new forecasting system. By introducing this bias, we are:
• increasing the error of all underforecasts of $\sigma - \sigma_n$ percents.
• lowering the average error of overforecasts (however the quantification is unclear).

Dismissing the improvement brought by the bias on overforecasts, we see that, in the worst case, the accuracy of the new - and now biased - forecasting system is degraded of $\sigma - \sigma_n$ percents, which turns into an overall accuracy that remains lower or equal to $\sigma$.

Then, we note that the total amount of inventory $V$ is proportional to the lead demand. The behavior is explicit when using a safety stock model for determining inventory levels, but basically, it applies for alternative methodologies as well.

By lowering the forecasts of $\sigma - \sigma_n$ percents, we are thus applying a similar reduction on the amount of inventory $V$. Then, since accuracy of the biased system remains lower to $\sigma$, the stockouts frequency should also stay lower than the one of the old system.

Finally, we have shown that based on a more accurate forecast, it is possible to build a lower inventory level of $\sigma - \sigma_n$ percents that does generate more stockouts - because forecasts remain better or equal (accuracy wise) to the ones of the old system.

Thus, the inventory reduction is $V \left(\sigma - \sigma_n \right)$. Considering the total yearly friction costs $H$, this reduction generates savings equal to $B=V H \left(\sigma - \sigma_n \right)$.

The variable $H$ should include all friction costs involved with the possession of inventory. In particular, a misconception that we routinely observe consists of stating that the value of $H$ is between 4% and 6%. However, that is only the cost for the company to fund its working capital by borrowing money to the bank.