Equation 6-15 provides the general formula for determining the reorder point. When demand during the lead time is normally distributed, the reorder point becomes

where

Thus, the amount of safety stock is simply $\text{Z}{\sigma }_{\text{d}\text{LT}}.$ The following example looks at how to determine the appropriate safety stock level when demand during the lead time is normally distributed and the mean and standard deviation are known.

The Hinsdale Company carries a variety of electronic inventory items, and these are typically identified by SKU. One particular item, SKU A3378, has a demand that is normally distributed during the lead time, with a mean of 350 units and a standard deviation of 10. Hinsdale wants to follow a policy that results in stockouts occurring only 5% of the time on any order. How much safety stock should be maintained, and what is the reorder point? Figure 6.8 helps visualize this example.

From the normal distribution table (Appendix A), we have $Z=1.65:$

So the reorder point is 366.5, and the safety stock is 16.5 units.

If the mean and standard deviation of demand during the lead time are not known, they must be calculated from historical demand and lead time data. Once these are found, Equation 6-16 can be used to find the safety stock and reorder point. Throughout this section, we assume that lead time is in days, although the same procedure can be applied to weeks, months, or any other time period. We will also assume that if demand fluctuates, the distribution of demand each day is identical to and independent of demand on other days. If both daily demand and lead time fluctuate, they are also assumed to be independent.

There are three situations to consider. In each of the following ROP formulas, the average demand during the lead time is the first term and the safety stock $\left(Z{\sigma }_{d\text{LT}}\right)$ is the second term.

1. Demand is variable but lead time is constant:

   $$\text{ROP}=\overline{d}L+Z\left({\sigma }_d\sqrt{L}\right)$$ (6-17)

   where

   $$\begin{array}{lll}\hfill \overline{d}& =\hfill & \text{average daily demand}\hfill \\ \hfill {\sigma }_d& =\hfill & \text{standard deviation of daily demand}\hfill \\ \hfill L& =\hfill & \text{lead time in days}\hfill \end{array}$$

2. Demand is constant, but lead time is variable:

   $$\text{ROP}=d\overline{L}+Z\left(d{\sigma }_L\right)$$ (6-18)

   where

   $$\begin{array}{lll}\hfill \overline{L}& =\hfill & \text{average}\text{lead time }\hfill \\ \hfill {\sigma }_L& =\hfill & \text{standard deviation of lead time}\hfill \\ \hfill d& =\hfill & \text{daily}\text{demand}\hfill \end{array}$$

3. Both demand and lead time are variable:

   $$\text{ROP}=\overline{d}\overline{L}+Z\left(\sqrt{\overline{L}{\sigma }_d^2+{\overline{d}}^2{\sigma }_L^2}\right)$$ (6-19)

Notice that the third situation is the most general case and the others can be derived from that. If either demand or lead time is constant, the standard deviation and variance for that would be 0, and the average would just equal the constant amount. Thus, the formula for ROP in situation 3 can be simplified to the ROP formula given for that situation.

Hinsdale has decided to determine the safety stock and ROP for three other items: SKU F5402, SKU B7319, and SKU F9004.

For SKU F5402, the daily demand is normally distributed, with a mean of 15 units and a standard deviation of 3. Lead time is exactly 4 days. Hinsdale wants to maintain a 97% service level. What is the reorder point, and how much safety stock should be carried?

From Appendix A, for a 97% service level $Z=\mathrm{1.88.}$ Since demand is variable but lead time is constant, we find

So the average demand during the lead time is 60, and the safety stock is 11.28 units.

For SKU B7319, the daily demand is constant at 25 units per day, and the lead time is normally distributed, with a mean of 6 days and a standard deviation of 3. Hinsdale wants to maintain a 98% service level on this particular product. What is the reorder point?

From Appendix A, for a 98% service level $Z=\mathrm{2.05.}$ Since demand is constant but lead time is variable, we find

So the average demand during the lead time is 150, and the safety stock is 154.03 units.

For SKU F9004, the daily demand is normally distributed, with a mean of 20 units and a standard deviation of 4, and the lead time is normally distributed, with a mean of 5 days and a standard deviation of 2. Hinsdale wants to maintain a 94% service level on this particular product. What is the reorder point?

From Appendix A, for a 94% service level $Z=\mathrm{1.55.}$ Since both demand and lead time are variable, we find

So the average demand during the lead time is 100, and the safety stock is 63.53 units.

As the service level increases, the safety stock increases at an increasing rate. Table 6.5 illustrates how the safety stock level would change in the Hinsdale Company (SKU A3378) example for changes in the service level. As the amount of safety stock increases, the annual holding cost increases as well.

When the EOQ assumptions of constant demand and constant lead time are met, the average inventory is $Q/2,$ and the annual holding cost is $\left(Q/2\right)C_h.$ When safety stock is carried because demand fluctuates, the holding cost for this safety stock is added to the holding cost of the regular inventory to get the total annual holding cost. If demand during the lead time is normally distributed and safety stock is used, the average inventory on the order quantity (Q) is still $Q/2,$ but the average amount of safety stock carried is simply the amount of the safety stock (SS) and not one-half this amount. Since demand during the lead time is normally distributed, there would be times when inventory usage during the lead time exceeded the expected amount and some safety stock would be used. But it is just as likely that the inventory usage during the lead time would be less than the expected amount and the order would arrive while some regular inventory remained in addition to all the safety stock. Thus, on the average, the company would always have this full amount of safety stock in inventory, and a holding cost would apply to all of this. From this, we have

Total annual holding cost $=$ Holding cost of regular inventory $+$ Holding cost of safety stock

where

In the Hinsdale example for SKU A3378, let’s assume that the holding cost is $2 per unit per year. The amount of safety stock needed to achieve various service levels is shown in Table 6.5. The holding cost for the safety stock would be these amounts times $2 per unit. As illustrated in Figure 6.9, this holding cost would increase extremely rapidly once the service level reached 98%.

To use Excel QM to determine the safety stock and reorder point, select Excel QM from the Add-Ins tab, and select Inventory—Reorder Point/Safety Stock (normal distribution). Enter a title when the input window appears, and click OK. Program 6.4A shows the input screen and formulas for the first three Hinsdale Company examples. Program 6.4B shows the solution.