Chapter 67: The Economic Order Quantity Inventory Model

Overview

• An electronics store sells 10,000 PDAs per year. Each time an order is placed for a supply of PDAs, the store incurs an order cost of $10. The store pays $100 for each PDA, and the cost of holding a PDA in inventory for a year is assumed to be $20. When the store orders PDAs, how large an order should it be?

• A manufacturing plant produces 10,000 computers per year. The cost to produce each computer is $2,000. The cost to set up a production run of computers is $200, and the cost to hold a computer in inventory for a year is $500. The plant can, if it wants, produce 25,000 computers per year. When the plant produces computers, how large a batch should it produce?

When a store orders an item repeatedly, a natural question is, what quantity should the store order each time? If the store orders too many items, it incurs excessive inventory or holding costs. If the store orders too few items, it incurs excessive re-ordering costs. Somewhere, there must be a happy medium that minimizes the sum of annual inventory and order costs.

Similarly, consider a manufacturing plant that produces batches of a product. What batch size minimizes the sum of annual inventory and setup costs? The two examples in this chapter show how to use the Economic Order Quantity formula (developed in 1913 by F. Harris of Westinghouse Corporation) to answer these questions.

• An electronics store sells 10,000 PDAs per year. Each time an order is placed for a supply of PDAs, an order cost of $10 is incurred. The store pays $100 for each PDA, and the cost of holding a PDA in inventory for a year is assumed to be $20. When the store orders PDAs, how large an order should it be?

• The size of an order that minimizes the sum of annual inventory and ordering costs can be determined after the following parameters are known:

  • K=Cost per order

  • h=Cost of holding one unit in inventory for a year

  • D=Annual demand for product

• You can follow an example of how to work with these parameters using the EOQ worksheet in the file Eoq.xlsx, which is shown in Figure 67-1.

  
  Figure 67-1: EOQ template

• If q equals order size, annual inventory cost equals 0.5qh. (Throughout this example, I’ll refer to this equation as Equation 1.) We derive Equation 1 because our average inventory level (0.5q) will be half the maximum inventory level. To see why the average inventory level is 0.5q, note that we can compute the average inventory level for a cycle (the time between the arrival of orders). At the beginning of a cycle, an order arrives, and our inventory level is q. At the end of the cycle, we are out of stock, and our inventory level is 0. Because demand occurs at a constant rate, the average inventory level during a cycle is simply the average of 0 and q or 0.5q. Maximum inventory level will equal q because orders are assumed to arrive at the instant that the inventory level is reduced to 0.

• Because D/q orders are placed per year, annual ordering cost equals (D/q)*K. (I’ll refer to this equation as Equation 2.) Using calculus or the Microsoft Office Excel 2007 Solver, we can show that the annual sum of inventory and ordering costs is minimized for a value of q equal to the economic order quantity (EOQ), which is calculated using the following formula. (I’ll refer to this equation as Equation 3.)

  

• From this formula, we see the following:

  • An increase in demand or ordering cost will increase the EOQ.

  • An increase in holding cost will decrease the EOQ.

• In the file Eoq.xlsx, I use Equation 3 to determine EOQ in cell C5. I determine annual holding cost in cell C6 by using Equation 1. I determine annual ordering cost in cell C7 with Equation 2. Notice that for EOQ, the annual ordering cost equals the annual holding cost, which will always be the case. In cell C8, I determine the total annual cost (ignoring the purchasing cost, which does not depend on our ordering strategy) with the formula C6+C7.

• Of course, you can use one-way and two-way data tables to determine the sensitivity of the EOQ and various costs to variations in K, h, and D. For our example, we have K=$10, D=10,000 PDAs per year, and h=$20 per PDA. Inserting these values in cells C2:C4, we find the following:

  • Each order should be for 100 PDAs.

  • Annual holding and ordering costs each equal $1,000. The EOQ always sets annual holding costs equal to annual ordering costs.

  • Total annual costs (exclusive of purchasing costs) equal $2,000.

• When you are working with EOQ, keep the following in mind:

  • The presence of quantity discounts invalidates the EOQ because the annual purchase cost then depends on the order size.

  • The EOQ assumes that demand occurs at a relatively constant rate throughout the year. The EOQ should not be used for products for which there is seasonal demand.

  • Annual holding cost is usually assumed to be between 10 percent and 40 percent of a product’s unit purchasing cost.

  • I’ve included (in the EOQ Protected worksheet in the file Eoq.xlsx) a version of the EOQ worksheet in which all formulas are protected. When the sheet is protected, nobody can change our formulas. See Chapter 65, “Pricing Stock Options,” for instructions about how to protect a worksheet.

  • For further discussion of inventory modeling, interested readers can refer to my book Operations Research: Applications and Algorithms (Duxbury Press, 2003).

• A manufacturing plant produces 10,000 computers per year. The cost to produce each computer is $2,000. The cost to set up a production run of computers is $200, and the cost to hold a computer in inventory for a year is $500. The plant can, if it wants, produce 25,000 computers per year. When the plant produces computers, how large a batch should it produce?

• With the EOQ model, we assume an order arrives the instant the order is placed. When a company manufactures a product instead of ordering it, an order must be produced and cannot arrive instantaneously. In such situations, instead of computing the cost-minimizing order quantity, we need to determine the cost-minimizing batch size. When a company produces a product internally instead of purchasing the product externally, the batch size that minimizes costs depends on the following parameters:

  • K=Cost of setting up a batch for production

  • h=Cost of holding each unit in inventory for a year

  • D=Annual demand for product

  • R=Annual rate at which the product can be produced. For example, IBM might have the capacity to produce 25,000 computers per year.

• If q equals the size of each production batch, the annual holding cost equals 0.5*(q/R)*(R–D)*h. (I’ll refer to this equation as Equation 4.) Equation 4 follows because each batch takes q/R years to produce and, during a production cycle, inventory increases at a rate of R–D. Our maximum inventory level, which occurs at the completion of a batch, can be calculated as (q/R)*(R–D). Our average inventory level will thus equal 0.5*(q/R)*(R–D).

• Because D/q batches are produced per year, annual setup cost equals KD/q (which I’ll refer to as Equation 5). Using calculus or the Excel Solver, we can show that the batch size that minimizes the sum of annual setup and production-run costs is given by the following (which I’ll refer to as Equation 6). We call this model the economic order batch (EOB) size.

  

• From this formula, we find the following:

  • An increase in K or D will increase the EOB.

  • An increase in h or R will decrease the EOB.

• In the Cont Rate EOQ worksheet in the file Contrateeoq.xlsx, I’ve constructed a template to determine the EOB, annual setup, and holding costs. The worksheet is shown in Figure 67-2.

  
  Figure 67-2: Template for computing EOB

• For our example, K=$200, h=$500, D=10,000 units per year, and R=25,000 units per year. After entering these parameter values in the cell range C2:C5, we find the following:

  • The batch size that minimizes costs is 115.47 computers. Thus, we should produce 115 or 116 computers in each batch.

  • The annual holding cost and setup costs equal $17,320.51. Again, the EOB will always set annual holding cost equal to annual setup cost.

  • Total annual cost (exclusive of variable production costs) is $34,641.02.

  • 86.6 batches per year will be produced.

• When you are working with the EOB model, keep the following in mind:

  • If the unit variable cost of producing a product depends on the batch size, the EOB model is invalid.

  • The EOB assumes that demand occurs at a relatively constant rate throughout the year. The EOB should not be used for products for which there is seasonal demand.

  • The annual holding cost is usually assumed to be between 10 percent and 40 percent of a product’s unit purchasing cost.

  • I’ve included (in the Protected worksheet in the file Contrateeoq.xlsx) a version of the EOB worksheet in which all formulas are protected. See Chapter 65 for instructions about how to protect a worksheet.