### Joannes Vermorel

The stock reward function is a key ingredient to make the most of probabilistic forecasts in order to boost your supply chain performance. The stock reward is used for computing the return on investment for every extra unit of stock to be purchased or manufactured.

The stock reward function is expressive and can be used like a mini-framework for addressing many different situations. However, as a minor downside, it’s not always easy to make sense of the calculations performed with the stock reward function. Below you’ll find a short list of graphs that represent the various transformations applied to the forecasts.

The first graph - entitled *Future demand* - represents a probabilistic demand forecast associated with a given SKU. The curve represents a distribution of probabilities, with the total area under the curve equal to one. In the background, this future demand is implicitly associated with a probabilistic lead time forecast, also represented as a distribution of probabilities. Such a distribution is typically generated through a probabilistic forecasting engine.

The *Marginal fill rate* graph represents the fraction of extra demand that is captured by each extra unit of stock. In other words, this graph demonstrates what happens to the fill rate as the stock increases. Since we are representing a marginal fill rate here, the total area under the curve remains equal to one. The marginal fill rate distribution can be computed with the fillrate() function.

The *Demand with backorders* graph is identical to the *Future demand* graph, except that 8 units have been introduced to represent a back order. The backorder represents guaranteed demand since these units have already been bought by clients. As a result, when backordered units are introduced, the probability distribution of demand is shifted to the right as the backordered units being guaranteed demand. The shift operator » is available as part of the algebra of distribution to compute such a transformation over the initial distribution.

The *Fill rate with backorders* graph is also very similar to the original *Marginal fill rate* graph, but has also been shifted 8 units to the right. Here, the plotted fill rate is only associated with the uncertain demand, hence the shape of the distribution remains the same.

The *Margin* graph represents the margin economic reward as computed by the stock reward function taking the *Demand with backorders* as input. The stock reward can be visualized as a distribution, but this is not a distribution of probabilities: the area under the curve is not equal to one but is instead equal to the total margin which would be captured with unlimited inventory. On the left of the graph, each backordered unit yields the same margin, which is not surprising as there is no uncertainty in capturing the margin given that the units have already been bought.

The *Stockout penalty* represents the second component of the stock reward function. The shape of the distribution might feel a bit unexpected, but this shape merely reflects that, by construction of the stock reward function, the total area under the curve is zero. Intuitively, starting from a stock level of zero, we have the sum of all the stockout penalties as we are missing all the demand. Then, as we move to the right with higher stock levels we are satisfying more and more demand and thus further reducing the stockout penalties; until there is no penalty left because the entire demand has been satisfied. The stock-out penalty of not serving backorders is represented as greater than the penalty of not serving the demand that follows. Here, we are illustrating the assumption that clients who have already backordered typically have greater service expectations than clients who haven’t yet bought any items.

The *Carrying costs* graph represents the third and last component of the stock reward function. As there is no upper limit for the carrying costs - it’s always possible to keep one more unit in stock thus further increasing the carrying costs - the distribution is divergent: it tends to negative infinity on the right. The total area under the curve is negative infinity, although this is a rather theoretical perspective. On the right, the carrying costs associated with the backordered units are zero: indeed, as those units have already been bought by clients they won’t incur any carrying costs, since those units will be shipped to clients as soon as possible.

The final stock reward - not represented above - would be obtained by summing the three components of the stock reward function. The resulting distribution would be interpreted as the ROI for each extra unit of stock to be acquired. This distribution typically starts with positive values,the first units of stock being profitable, but converge to negative infinity as we move to higher stock levels given the unbounded carrying costs.

The term support (mathematics) classically refers to the demand levels associated with non-zero probabilities. In the graphs above, the term *support* is used loosely to refer to the entire range that needs to processed as non-zero values by Envision. In particular, it’s worth mentioning that there are multiple calculations that require the distribution support to be extended in order to make sure that the final resulting distribution isn’t truncated.

- The shift operation, which happens when backorders are present, requires the support to be increased by the number of backordered units.
- The margin and carrying cost components of the stock reward function have no theoretical limits on the right, and can require arbitrarily large extensions of the support.
- Ordering constraints, such as MOQs, may require having inventory levels that are even greater than the ones reached by the shifted distributions. Properly assessing the tail of the distribution is key for estimating whether the MOQ can be profitably satisfied or not.

In practice, the Envision runtime takes care of automatically adjusting the support to make sure that distributions aren’t truncated during the calculations.