Tools: The Reorder Point (R, Q) Model (2024)

The basic economic order quantity (EOQ) model is a simple but effective tool to illustrate and optimize the tradeoff between ordering and holding costs. We call it simple because we assume away all demand and lead time uncertainty. In the EOQ model, we don't even consider stockout costs. The good news: We can build on the EOQ model to incorporate uncertainty and its influence on cost and inventory decisions. Let's do that now by learning about what is widely known as the reorder point model (see Figure 9-4). It is also referred to as the R, Q model because it is defined by the reorder point (R) and the order quantity (Q). The reorder point model helps you decide when replenishment orders should be placed (reorder point) and how large such orders should be (order quantity). Table 9-1 highlights the key similarities and differences between the basic EOQ and reorder point models.

A Conceptual Overview of the Reorder Point Model

Figure 9-5 illustrates how the reorder point model works. Inventory levels are plotted on the vertical axis and time on the horizontal axis. Let's return to the example where you were a textile merchant ordering skirts from a supplier. Look at the left-hand side of the graph. To begin with, you hold a certain number of skirts in inventory, which you sell to customers over time. Demand is random, so, your inventory decreases in a non-linear fashion (i.e., the line is not straight). Since you have an order fulfillment lead time, you can't simply wait to place your order until all of your inventory is depleted. If you did wait, you would suffer stockouts—making your customers mad. You need to place your replenishment orders well before you run out of inventory. The inventory level at which you place your order is called the reorder point ( R).

Again, look at the left-hand side of the graph. Let's break the reorder point into its two components: average lead time demand and safety stock. Now, imagine you decide not to carry any safety stock (i.e., safety stock is zero). You would place replenishment orders when there is just enough inventory left to cover average demand during the average lead time. What is the probability you would stock out in this scenario? You can see this by looking at the flipped distribution of lead time demand shown on the right-hand side of the graph. The dotted line hits the distribution at the midpoint. In other words, the likelihood of stocking out is 50%.

Clearly, you don't want to stock out in half of all order cycles. After all, you wouldn't shop at a store that was out of the products you came in to buy 50% of the time? So, if you want to reduce the stockout probability (highlighted in red in Figure 9-5) and achieve greater customer service levels, you need to carry safety stock. Adding safety stock increases three things:

1. Your reorder point.

2. Your total inventory levels.

3. Your carrying costs.

You need to choose a safety stock level that balances inventory holding and stockout costs. Finally, when building your reorder point model, you need to include order placements costs—just like in the basic EOQ model.

Determining Optimal Reorder Points and Order Quantities

The first step in identifying optimal inventory decisions is to set up the total cost function. What do you think this cost function looks like? Think back to Figure 9-6. Your total costs vary as a function of your order quantity Q and your safety stock, which is a component of the reorder point R. You can picture this cost function as approximately bowl-shaped as illustrated in Figure 9-4. The lowest total cost solution is obtained when you simultaneously choose the optimal order quantity Q* and the optimal reorder point R*.

To develop the formula needed to calculate the optimal (i.e., lowest) costs, you need to build and then optimize your total cost function. The total cost function will include the following four cost categories:

• Cycle stock holding costs (h in $ per year)

• Safety stock holding costs (h in $ per year)

• Order placement costs (K in $)

• Stockout costs (p in $ per unit)

The math for this total cost function is shown below. In this function, D is annual demand (in units), Q represents the order quantity (in units), SS stands for safety stock (in units), and n(R) is the stockout quantity per order cycle.

Your total costs are the sum of the following:

• Total inventory holding costs, which are calculated as the sum of cycle stock (Q/2) and safety stock (SS) multiplied by the holding cost per unit per year (h).

• Order placement costs are the total number of orders placed per year ( D/Q) multiplied by the cost of placing a single order ( K).

• Annual stockout costs are obtained by multiplying the unit stockout cost ( p) with the number of units out-of-stock per cycle [n(R)] and then multiplying this product by the number of cycles per year ( D/Q).

When you optimize the above cost function and solve for order quantity and reorder point, you get the following formulas:

$$Q^{\ast }=\sqrt{\frac{2\cdot D\cdot (K+p\cdot n(R))}{h}}andF(R^{\ast })=1-\frac{Q\cdot h}{p\cdot D}$$

Notice that the optimal order quantity is similar to the basic EOQ. The only difference is that stockout costs per cycle are added as part of your tradeoff considerations. F(R*), in turn, identifies the optimal in-stock probability, which reflects the balance between holding and stockout costs. This in-stock probability defines the optimal level of safety stock and the reorder point as illustrated in Figure 9-7. Now, let's make one point clear: The two equations (Q* and F(R*)) are interdependent. In other words, the equation for Q* includes R and the equation for R* contains Q. This means you can't solve the equations independently. You need to use an iterative approach—a trick that is hard to do by calculator, but easily handled by a spreadsheet.

Now, what would make you want to hold more safety stock? One reason: Your risk as measured by the standard deviation of lead time demand ( sLD) is high. Another possibility: Your cost structure suggests a higher optimal in-stock probability,F(R*). Simply put, some mix of your exposure to risk and the optimal service level influences how much safety stock you should hold. Let's express this in a formula: SS=z∙sLD, where z is a mathematical function of the optimal service level. Your reorder point is the combination of average lead time demand ( LTD) and safety stock ( SS). The formula is:

$$R=LTD+SSorR=LTD+z\ast sLD$$

We admit that all these distributions and formulas may be a bit intimidating at first but the good news: Microsoft Excel can do the math for you. Now, let's practice. Imagine you are the inventory manager for OrGranola, a manufacturer of high-end, all-natural granola bars. OrGranola distributes its top-selling product, "Taste of Heaven" through its own DC to stores in Arizona, New Mexico and Utah. You obtain the following information from the DC:

Given this information, what are the DC's optimal reorder point and order quantity? What are the total annual inventory-related costs? Watch the video to see exactly how this works.

The Reorder Points Model for Unknown Stockout Costs

One drawback to using the optimal reorder point model is that many managers have no idea what a stockout actually costs. After all, stockout costs are hard to calculate. You need to know how all of your different customers really respond to stockout situations. Some simply switch SKUs (buying a larger or smaller package size) or switch brands. Others postpone the buying decision. A few customers may be so ticked off that they never buy from you again. The data you need to calculate the cost of a stockout is very, very difficult to get. As a result, only a small percent of companies have accurate stockout costs. 1 So, as a bright, talented replenishment manager who wants to help your company better manage inventory by defining optimal order quantities and optimal reorder points, what do you do? You can't use the R,Q model because it requires stockout costs.

If you find yourself in a situation where you don't know stockout costs, the best you can do is use the basic EOQ and managerially define an in-stock target. This is called a "managerial" reorder point model:

In practice, you'll find that firms often set in-stock targets, often ranging somewhere between 90% and 99%. Unfortunately, if you don't know the stockout costs, you really cannot ascertain whether such targets are optimal or even near optimal. Implementing a managerial reorder point policy is fairly straightforward. Let's work through an example: Once again, imagine you are the inventory manager for OrGranola and use the same scenario—except without stockout costs. If you target a 96% in-stock rate, what is your optimal order quantity and reorder point? At this point, just plug the numbers from the table into your two equations (Q, R). If you want to see how to do this in an Excel spreadsheet, watch the video: