The appropriate level of safety inventory is determined by the following two factors:

- The uncertainty of both demand and supply
- The desired level of product availability

As the uncertainty of supply or demand grows, the required level of safety inventories increases. Demand for milk at a supermarket is quite predictable. As a result, supermarkets can operate with low levels of safety inventory relative to demand. In contrast, demand for spices at the same supermarket is much harder to predict. Thus the supermarket needs to carry high levels of safety inventory for spices relative to demand. Whereas most of the milk inventory at a supermarket is cycle inventory (with very little being safety inventory), most of the spice inventory is safety inventory carried to deal with uncertainty of demand.

As the desired level of product availability increases, the required level of safety inventory also increases. If the supermarket targets a higher level of product availability for a certain spice, it must carry a higher level of safety inventory for that spice.

Next, we discuss some measures of demand uncertainty.

**1. Measuring Demand Uncertainty**

As discussed in Chapter 7, demand has a systematic as well as a random component. The random component is a measure of demand uncertainty. The goal of forecasting is to predict the systematic component and estimate the random component. The random component is usually estimated as the standard deviation of forecast error. We illustrate our ideas using uncertain demand for a smartphone at B&M Office Supplies as the context. We assume that periodic demand for the phone at B&M is normally distributed with the following inputs:

D: Average demand per period

s_{D}: Standard deviation of demand (forecast error) per period

Even though standard deviation of demand is not necessarily the same as forecast error, we treat the two to be interchangeable in our discussion. Safety inventory calculations should really be based on forecast error.

Lead time is the gap between the time an order is placed and when it is received. In our discussion, we denote the lead time by L. In the B&M example, L is the time between when B&M orders phones and when they are delivered. In this case, B&M is exposed to the uncertainty of demand during the lead time. Whether B&M is able to satisfy all demand from inventory depends on the demand for phones experienced during the lead time and the inventory B&M has when a replenishment order is placed. Thus, B&M must estimate the uncertainty of demand during the lead time, not just in a single period. We now evaluate the distribution of demand over L periods, given the distribution of demand during each period.

**EVALUATING DEMAND DISTRIBUTION OVER L PERIODS** Assume that demand for each period i, i = 1, . . . , L, is normally distributed with a mean D_{t} and standard deviation s_{i}. Let p_{i}j be the correlation coefficient of demand between periods i and j. In this case, the total demand during L periods is normally distributed with a mean of D_{L} and a standard deviation of s_{L}, where the following is true:

Demand in two periods is perfectly positively correlated if rij = 1. Demand in two periods is perfectly negatively correlated if rij = -1. Demand in two periods is independent if rij = 0. If demand during each of L periods is independent and normally distributed with a mean of D and a standard deviation of sD, Equation 12.1 can be used to show that total demand during the L periods is normally distributed with a mean DL and a standard deviation of sL, where the following is true:

Another important measure of uncertainty is the coefficient of variation (cv), which is the ratio of the standard deviation to the mean. Given demand with a mean of m and a standard deviation of s, we have

The coefficient of variation measures the size of the uncertainty relative to demand. It captures the fact that a product with a mean demand of 100 and a standard deviation of 100 has greater demand uncertainty than a product with a mean demand of 1,000 and a standard deviation of 100. Considering the standard deviation alone cannot capture this difference.

Next, we discuss some measures of product availability.

**2. Measuring Product Availability**

Product availability reflects a firm’s ability to fill a customer order out of available inventory. A stockout results if a customer order arrives when product is not available. There are several ways to measure product availability. Some of the important measures are listed next.

*Product fill rate (fr)*is the fraction of product demand that is satisfied from product in inventory. Fill rate is equivalent to the probability that product demand is supplied from available inventory. Fill rate should be measured over specified amounts of demand rather than over time. Thus, it is more appropriate to measure fill rate over every million units of demand rather than every month. Assume that B&M provides smartphones to 90 percent of its customers from inventory, with the remaining 10 percent lost to a neighboring competitor because of a lack of available inventory. In this case, B&M achieves a fill rate of 90 percent.*Order fill rate*is the fraction of orders that are filled from available inventory. Order fill rate should also be measured over a specified number of orders rather than over time. In a multiproduct scenario, an order is filled from inventory only if all products in the order can be supplied from the available inventory. In the case of B&M, a customer may order a phone along with a laptop. The order is filled from inventory only if both the phone and the laptop are available through the store. Order fill rates tend to be lower than product fill rates because all products must be in stock for an order to be filled.*Cycle service level (CSL)*is the fraction of replenishment cycles that end with all the customer demand being met. A replenishment cycle is the interval between two successive replenishment deliveries. The CSL is equal to the probability of not having a stockout in a replenishment cycle. CSL should be measured over a specified number of replenishment cycles. If B&M orders replenishment lots of 600 phones, the interval between the arrival of two successive replenishment lots is a replenishment cycle. If the manager at B&M manages inventory such that the store does not run out of inventory in 6 out of 10 replenishment cycles, the store achieves a CSL of 0.6 or 60 percent. Observe that a CSL of 0.6 typically results in a much higher fill rate. In the 60 percent of cycles in which B&M does not run out of inventory, all customer demand is satisfied from available inventory. In the 40 percent of cycles in which a stockout does occur, most of the customer demand is satisfied from inventory. Only the small fraction toward the end of the cycle that arrives after B&M is out of inventory is lost. As a result, the fill rate is much higher than 0.6.

The distinction between product fill rate and order fill rate is usually not significant in a single-product situation. When a firm is selling multiple products, however, this difference may be significant. For example, if most orders include 10 or more products that are to be shipped, an out-of-stock situation of one product results in the order not being filled from stock. The firm in this case may have a poor order fill rate even though it has good product fill rates. Tracking order fill rates is important when customers place a high value on the entire order being filled at one time.

Next, we describe two replenishment policies that are often used in practice.

**3. Replenishment Policies**

A replenishment policy consists of decisions regarding when to reorder and how much to reorder. These decisions determine the cycle and safety inventories along with the fill rate fr and the cycle service level CSL. Replenishment policies may take any of several forms. We restrict attention to two types:

*Continuous review:*Inventory is continuously tracked, and an order for a lot size Q is placed when the inventory declines to the reorder point (ROP). As an example, consider the store manager at B&M who continuously tracks the inventory of phones. She orders 600 phones when the inventory drops below ROP = 400. In this case, the size of the order does not change from one order to the next. The time between orders may fluctuate, given variable demand.

*Periodic review*: Inventory status is checked at regular periodic intervals, and an order is placed to raise the inventory level to a specified threshold. As an example, consider the purchase of flash drives at B&M. The store manager does not track flash drive inventory continuously. Every Thursday, employees check flash drive inventory, and the manager orders enough so that the total of the available inventory and the size of the order equals 1,000 flash drives. In this case, the time between orders is fixed. The size of each order, however, can fluctuate given variable demand.

These inventory policies are not comprehensive, but they suffice to illustrate the key managerial issues concerning safety inventories.

Source: Chopra Sunil, Meindl Peter (2014), *Supply Chain Management: Strategy, Planning, and Operation*, Pearson; 6th edition.

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