This is part 4 of a 4-part series on inventory sawtooth curves.

Designing a truly reliable replenishment plan requires the ability to predict average (middle), maximum (top), and minimum (bottom) on-hand balances (OHBs) with accuracy. Sawtooth curves are a fantastic tool for exploring the differences between various replenishment plans in a visual way. However, graphing and visually analyzing every unique inventory item’s sawtooth curve, is often impractical if not outright impossible. To overcome this limitation, we need formulas.

## Basic Inventory Balance Formulas

Let’s look at an example of a sawtooth curve with steady demand and predicted minimum inventory.

Order Quantity |
1,200 |

Safety Stock |
200 |

Daily Demand |
100 |

LT Days |
10 |

Table 10. Steady Demand Sawtooth Curve

Figure 10. Steady Demand Sawtooth Curve

**Estimated Minimum Inventory**

For any replenishment plan, safety stock (SS) is equal to the expected minimum OHB. Since the minimum inventory is more than zero, there is safety stock (SS). A safety stock of 200 says that our minimum inventory balance is expected to be 200.

To estimate minimum inventory balance, determine what you need for one lead-time period by multiplying lead time days by daily demand. Next, subtract that result from your order quantity.

**Expected Minimum Inventory**= Safety Stock = Total order quantity during lead-time period – (Lead Time x Daily Demand)

Looking at our example, our order quantity is 1,200 and our daily demand is 1,000. The difference between the two is 200 units of safety stock.

**Expected Minimum Inventory**= 1,200 – 1,000 = 200 safety stock

**Estimated Average Inventory**

As you may recall from a previous article, average inventory is found at the vertical center of the sawtooth curve. Many managers find it intuitive to estimate average inventory by simply dividing the order amount in half. Unfortunately, this is misguided, resulting in under estimating average inventory.

To understand why, remember that we want to estimate average inventory, not average order quantity. Simply dividing order quantity in half fails to account for any SS, which always sits below an item’s order quantity.

**Estimated Average Inventory**= (Order Quantity / 2) + Safety Stock

In our example, dividing our order quantity of 1,200 by 2 gives us 600. Adding our safety stock of 200 units gives us an estimated average inventory of 800 units.

**Estimated Average Inventory**= (1,200 / 2) + 200 = 600

**Estimated Maximum Inventory**

To find estimated maximum inventory, is a simple function adding order quantity to safety stock.

**Estimated Maximum Inventory**= Order quantity + Safety Stock

In our example, estimated maximum inventory is 1,400.

**Estimated Maximum Inventory**= 1,200 + 200 = 1,400

**Supplier Imposed Conditions**

A primary inventory management goal is minimizing on-hand inventory in order to free up cash. Thus, its vital to recognize key 2 factors that often increase average inventory beyond what is otherwise mathematically sensible.

**High Minimum Order Quantity**

Ideal replenishment plans seek to order either exactly what can be consumed in a lead-time period or only slightly more than that. Ordering significantly more inventory beyond the recommended amount should be avoided.

Sometimes, artificially high order requirements are forced on us by our suppliers in the form of minimum order quantities or standard package quantities. When this is the case, the result is an extremely tall sawtooth curve, tied up cash, and reduced storage space availability.

Consider what would happen if our supplier required us to order a minimum of 8,000 units. Sticking with our previous example of daily demand of 100 units, the supplier’s minimum order quantity would cover a whopping 16 weeks!

When a supplier requires orders far larger than needed you only have 2 options available to remedy the situation:

- Convince the supplier to change its minimum order requirement.
- Switch suppliers.

Notice, that consignment is not one of the options presented here. That’s because although a properly designed consignment program can mitigate the negative effects of overstock on cash flow, no amount of consignment mitigates the effects of overstock on storage. Storage effects include but aren’t limited to lost production capacity due to reduced floor space, increased material handling costs due carrying more units, and much more. Never forget to account for storage in your replenishment plans.

**Lead Time **

Long lead times are another factor that negatively impact sawtooth curves. Recall, each peak on the sawtooth represents order replenishment. The distances between peaks are lead-time periods and the height of each peak equals the replenishment quantity plus safety stock.

When a peak’s height isn’t tall enough, this means there isn’t enough inventory to cover the demand. When orders must compensate for longer durations between receipts, one solution is to place more than one order per lead-time period.

## Multiple Orders per Lead Time

So far we have assumed that order quantity is determined by the quantity of units used during a single lead-time period (i.e. lead time demand). That is, we assumed that one order covered our entire lead time needs. However, this is only true for the simplest replenishment plan.

**Order quantity**= (Lead Time x Daily Demand) + SS

While simple is good, optimal is preferred, and often worth the marginal amount of added complexity. Consider, what would happen to our replenishment plan’s sawtooth curve if we placed multiple orders per lead time? A “simple” one-order per lead time replenishment plan has relatively fewer peaks, taller peaks, and longer distances between peaks. By contrast, when multiple orders are placed within a single lead-time period, the result is more peaks, shorter peaks, and less distance between those peaks.

Who cares about “peaks”? Well, said another way, placing multiple orders per lead time improves cash flow by reducing both maximum and average inventory levels. Let’s take a closer look…

**One Order Per Lead Time**

Consider the following replenishment scenario, summarized in Table 11. The item has a consistent daily demand of 50 units and a lead time of 20 days. Placing one order per 20-day lead time period gives us a lead time demand of 1,000 units. Thus, adding 50 units of safety stock to total lead time demand gives us an order quantity of 1,050. Figure 2 illustrates this replenishment plan over two lead time periods.

Order Quantity |
1,050 |

Safety Stock |
50 |

Daily Demand |
50 |

LT Days |
20 |

LT Demand |
1,000 |

LT Orders |
1 |

Table 11. 1 Orders Per Lead Time Period

Figure 11. 1 Order Per Lead Time Period

In this example, due the steady consistent consumption of this specific scenario, max inventory equals 1,050 units (total lead time demand + safety stock).

To find our Estimated Average Inventory we use our formula:

**Estimated Average Inventory**= Order Quantity / 2 + Safety Stock

**Estimated Average Inventory**= (1,050 / 2) + 50 = 575

Note, for purposes of illustration, the charts in the following examples plot beginning OHB, rather than ending OHB. However, this decision makes no difference in how we calculate max inventory or estimated average inventory.

**Two Orders Per Lead Time**

Now let’s consider what would happen if we were able to place two orders per lead time period, rather than only one, as seen in Table 12.

Order Quantity |
525 |

Safety Stock |
50 |

Daily Demand |
50 |

LT Days |
20 |

LT Demand |
1,000 |

LT Orders |
2 |

Table 12. 2 Orders Per Lead Time Period

Because we normally order 1,050 pieces for one lead-time period, we’ll need to divide that by 2 when placing two orders in a single lead-time period. So, we need to modify our order quantity calculation accordingly…

**= ((Lead Time x Daily Demand) + SS) / # of receipts per lead time.**

*Modified*Order quantityNote, that while we’ve “changed” our initial order quantity formula, this has no effect on our initial order quantity calculation for which essentially assumed only one receipt per lead time.

**Order Quantity**= 1,050 / 2 = 525

This should be straight forward. Because we’re doubling the number of orders needed for the same lead time demand, we’re simply dividing our total lead time demand and safety stock quantity in half.

Just as we have cut our order quantity in half by doubling the number of receipts per lead time, our replenishment time will be cut in half as well from being equal to our lead time of 20 days to 10 days.

What about average inventory?

**Estimated Average Inventory**= Order Quantity / 2 + SS

**Estimated Average Inventory**= (525 / 2) + 50 = 313

Our Estimated Average Inventory is 313.

Figure 12 illustrates the effect of placing 2 orders per lead time period.

Figure 12. 2 Orders Per Lead Time Period

The key here is to recognize the impact that placing 2 orders per lead time has on both maximum and average inventory levels. We’ve cut max inventory by 50% and average inventory by 46%!

Can we cut them even further?

What happens if we were to place and receive four orders per lead-time period?

Let’s find out!

**Four Replenishment Orders per Lead Time Period**

Recalling our modified formula for order quantity, since we normally order 1,050 pieces when placing one order per lead-time period, we’ll need to divide that by 4 when placing four orders in a single lead-time period.

Order Quantity |
263 |

Safety Stock |
50 |

Daily Demand |
50 |

LT Days |
20 |

LT Demand |
1,000 |

LT Orders |
4 |

Table 13. 4 Orders Per Lead Time Period

**Order Quantity**= 1,050 / 4 = 262.5 pieces

Next, we use the Average Inventory formula to find Estimated Average Inventory.

**Estimated Average Inventory** = (Order Quantity / 2) + SS

**Estimated Average Inventory**= 263 / 2 = 131.5 + 50 = 181.5

In this example, the Estimated Average Inventory is 182 (rounded up from 181.5)

Note, many people are tempted to alter the Estimated Average Inventory formula by dividing the order quantity by 4 instead of 2, basing it on the number of orders to be placed within a single lead-time period. Don’t do that. You have already accounted for the number of orders in your order quantity. Harmonizing the performance of deliveries and consumption of inventories depends on the accuracy of your replenishment plan and mistakes such as this can produce grossly inaccurate estimates.

Figure 13. 4 Orders Per Lead Time Period

So how does 4 orders per lead time compare to 2 orders? Well beyond simply having more peaks and shorter peaks, we’ve managed to reduce our max inventory by 75% and average inventory by a whopping 68%, simply by increasing the number of receipts!

Having gone through this exercise, it’s vital to be clear of this key takeaway: Increasing the number of replenishment orders per lead-time period can be an extremely effective method for reducing maximum and average inventory levels, and significantly increasing cash flow!

That said, why stop at 4 orders? Why not place 1 order per day for a total of 20 orders per lead time period? Better yet, why not replenish hourly if we really wish to drive inventory levels to their absolute minimum? Talk about a flat sawtooth curve! (By the way, this is sometimes very feasible. Falcon has more than one client that it replenishes on an hourly or more frequent basis.)

To answer our question; it depends. For one, it depends on whether the item is purchased or manufactured. For purchased items it depends on a given supplier’s capabilities, as well as the specific costs for your company to place orders and receive them. Just think of all the tasks that might be involved in the receiving process such as unloading, inspecting, and putting away, etc. For manufactured items, replenishment frequency depends on work schedules, set-up times, number of manufacturing processes, and various other internal operational constraints.

Ultimately, we need a framework capable of helping us to determine the appropriate replenishment quantity and frequency for any given inventory item. While seemingly simple, this determination can be extremely nuanced, and complicated especially when analyzing large numbers of inventory items. Fortunately, we have such as framework; kanban. In fact, **kanban** was built exactly for this purpose.

**Summary**

It’s unrealistic to create a sawtooth curve for every internal and external item. However, by using a few simple formulas, you can estimate minimum (bottom), average (middle) and maximum (top) inventory for every inventory item, quickly and easily.

That concludes part 4 of this series on inventory sawtooth curves. Should you have any questions about or recommendations for improving this or any of Falcon’s content don’t hesitate to let us know!