# Incorporating Safety Stock Decisions in Supply Chain Network Models

We know that single period Strategic Supply Chain Network models are kind of snapshots whereas Inventory decisions have dynamic effects which make it challenging to capture inventory planning phenomenon within a single period Network model. Even if the model is a multi-period model, the length of the periods will be long enough, generally a year or longer, to ignore within period variations in inventory. Hence, network models require some level of modeling artistry to capture inventory planning phenomenon.

I will focus on the challenge of modeling Safety Stock (SS) Inventory in a Network Optimization model in this post.

Why is incorporating Safety Stock in a Network Model challenging?

We know that Safety Stock (SS) for any product, say i, held at any DC depends on two key parameters-Expected Demand and the Standard Deviation of Demand. We also know that the relationships between these quantities and the Optimal Safety stock is non-linear and implicit functions of probability distribution. These aspects make it a difficult exercise in optimization modeling to explicitly model safety stock decisions and costs based on SS equations.

How can we incorporate Safety Stock in a Network Model?

There are two good approximation methods that can be used to capture SS in a Network Model.

Method 1

This method utilizes the classic principle of risk pooling methodology. In this approximation, you start by computing the safety stock cost (say SS1i) for a product i, assuming that all of that product is being supplied by one DC. You then project the cost to multiple DCs using the multiplier √N (where N is the number of DCs this product is assigned to). This approximation assumes that, as the optimization model assigns market demand for product k to various combinations of distribution centers, the statistical characteristics of uncertain demand handled by each is identical or very similar. This is a major assumption because some distribution centers will handle significantly larger demand than others.

Method 2

The second approximation method is to model SS cost for each product for each DC as a cost relationship with decreasing marginal cost, as shown in the graph below. The assumption underlying this curve is that all markets for a given product experience a constant rate of demand variability characterized by square (σ ) which is Variance per unit of expected yearly demand for the product. This implies that if the market has expected demand E(D) for this product, the variance of demand is square (σ ) x E(D).

Using this method, we can calculate Annual SS cost for different levels of flow, as show below. What we are doing basically is computing SS cost as a function of outflow at a particular DC for various levels of flows. This piecewise function can be then implemented in the Network model so that the model can calculate SS cost in a Network Optimization model.