The Economic-Order-Quantity Decision Model

Learning Objective 2 Balance ordering costs with carrying costs using the economic-order-quantity (EOQ) decision model

choose the inventory quantity per order to minimize these costs

How much should a firm order of a given product? The economic order quantity (EOQ) is a decision model that calculates the optimal quantity of inventory to order, given the following set of assumptions:

Under the simplest version of the EOQ model, there are only ordering and carrying costs, the most common costs of inventory.
The same quantity is ordered at each reorder point.
Demand, ordering costs, and carrying costs are known with certainty. The purchase-order lead time, the time between placing an order and its delivery, is also known with certainty.
The purchasing cost per unit is unaffected by the order quantity. This assumption makes purchasing costs irrelevant to determining the EOQ because the purchase price is the same, whatever the order size.
No stockouts occur. The basis for this assumption is that the costs of stockouts are so high that managers maintain adequate inventory to prevent them.
When deciding on the size of a purchase order, managers consider the costs of quality and shrinkage costs only to the extent that these costs affect ordering or carrying costs.

Note that EOQ analysis ignores purchasing costs, stockout costs, costs of quality, and shrinkage costs. Also recall from Chapter 12 that managers only consider relevant costs when making decisions. In a later section, we will discuss how to identify the relevant ordering and carrying costs. At this point, we simply note that EOQ is the order quantity that minimizes the sum of a company’s relevant ordering and carrying costs. The sum of the costs is the firm’s relevant total ordering and carrying cost of inventory. The relevant total costs are calculated as follows:

$Relevant total costs = Relevant ordering costs + Relevant carrying costs$

We use the following notations:

$\begin{array}{l} D = Demand in units for a specified period (one year in this example) \\ Q = Size of each order (order quantity) \end{array}$

$Number of purchase orders per period (one year) = \frac{Demand in units for a period (one year)}{Size of each order (order quantity)} = \frac{D}{Q}$

Average inventory in $units = \frac{Q}{2},$ because each time the inventory goes down to 0, an order for Q units is received. The inventory varies from Q to 0, so the average inventory is $\frac{0 + Q}{2} .$

$\begin{array}{l} P = Relevant ordering cost per purchase order \\ C = Relevant carrying cost of one unit in stock for the time period used for D (one year) \end{array}$

For any order quantity, Q,

$\begin{array}{l} Annual relevant ordering costs & = & (\begin{matrix} Number of \\ purchase orders \\ per year \end{matrix} \times \begin{matrix} Relevant ordering \\ cost per \\ purchase order \end{matrix}) = (\frac{D}{Q} \times P) \\ Annual relevant carrying costs & = & (\begin{matrix} Average inventory \\ in units \end{matrix} \times \begin{matrix} Annual \\ relevant carrying \\ cost per unit \end{matrix}) = (\frac{Q}{2} \times C) \\ Annual relevant total costs & = & \begin{matrix} Annual \\ relevant ordering \\ costs \end{matrix} + \begin{matrix} Annual \\ relevant carrying \\ costs \end{matrix} = (\frac{D}{Q} \times P) + (\frac{Q}{2} \times C) \end{array}$

The order quantity that minimizes annual relevant total costs is

$E O Q = \sqrt{\frac{2 D P}{C}}$

The EOQ model is solved using calculus, but the key intuition is that relevant total costs are minimized when relevant ordering costs equal relevant carrying costs. If carrying costs are lower (greater) than ordering costs, the total costs can be reduced by increasing (decreasing) the order quantity. To solve for EOQ, we set

$\begin{matrix} (\frac{Q}{2} \times C) = (\frac{D}{Q} \times P) \\ Multiplying both sides by \frac{2 Q}{C}, we get Q^{2} = \frac{2 D P}{C} \\ Q = \sqrt{\frac{2 D P}{C}} \end{matrix}$

The formula indicates that EOQ increases with higher demand and/or higher ordering costs and decreases with higher carrying costs.

Let’s see how EOQ analysis works. Glare Shade sells sunglasses. This problem focuses on Glare Shade’s basic sunglasses, UX1. Glare Shade purchases the UX1s from Rytek at $14 a unit. Rytek pays for all incoming freight. No inspection is necessary at Glare Shade because Rytek supplies quality merchandise. Glare Shade’s annual demand is 13,000 units of UX1s, at a rate of 250 units per week. Glare Shade requires a 15% annual rate of return on its investment. Relevant ordering cost per purchase order is $200.

The relevant carrying cost per unit per year is as follows:

A table shows relevant carrying cost per unit per year for Glare Shade

The table depicts relevant carrying cost per unit per year for Glare Shade. The details of the table are as follows:

1. Required annual return on investment, 0.15 times 14 dollars: 2.10 dollars

2. Relevant costs of insurance, materials handling, breakage, shrinkage, and so on, per year: 3.10 dollars

3. Total: 5.20 dollars

What is the EOQ for ordering UX1 sunglasses?

Substituting $begin inline style D equals 13 comma back space 000 thin space text units?per?year end text comma end style$ $inline style begin inline style P equals straight $ 200 thin space text per?order end text comma end style$ and $inline style begin inline style C equals straight $ 5.20 thin space text per?unit end text end style$ per year, in the EOQ formula, we get

$E O Q = \sqrt{\frac{2 \times 13,000 \times 200}{$ 5.20}} = \sqrt{1,000,000} = 1,000 units$

Purchasing 1,000 units per order minimizes total relevant ordering and carrying costs. Therefore, the number of deliveries each period (1 year in this example) is as follows:

$\frac{D}{E O Q} = \frac{13,000}{1,000} = 13 deliveries$

Recall the annual relevant total costs $(RTC) = (\frac{D}{Q} \times P) + (\frac{Q}{2} \times C)$

For $inline style Q equals text 1,000 end text thin space text units end text comma$

$\begin{array}{l} R T C & = & \frac{13,000 \times $ 200}{1,000} + \frac{1,000 \times $ 5.20}{2} \\ = & $ 2,600 + $ 2,600 = $ 5,200 \end{array}$

Exhibit 21-1 graphs the annual relevant total costs of ordering $(D P / Q)$ and carrying inventory $open parentheses Q C divided by 2 close parentheses$ under various order sizes (Q), and it illustrates the tradeoff between these two types of costs. The larger the order quantity, the lower the annual relevant ordering costs but the higher the annual relevant carrying costs. The annual relevant total costs are at a minimum at the EOQ at which the relevant ordering and carrying costs are equal.

Exhibit shows graph depicting the analysis of ordering costs and carrying costs for UX1 sunglasses at Glare Shade.

Details of the graph is as follows: Y-axis shows "Order Quantity (Q)" in units ranging from 0 to 2600 with increments of 650. X-axis shows "Relevant Total Costs" in dollars ranging from 0 to 10000 with increments of 2000. The graph shows three lines. A curved line, depicting "Annual relevant carrying costs in (QC over 2)" is drawn from the origin, rising with an upward slope. Another curve, depicting "Annual relevant ordering costs (DP over Q)" is drawn with a steep downward slope. Another U-shaped curve, depicting "Annual relevant total costs" is drawn. The minimum point of this curve is marked at a coordinate (1000, 5200). Value X = 1000 is labeled as "EOQ = 1000".

When to Order, Assuming Certainty

The second decision that Glare Shade’s managers face is when to order the units. The reorder point is the quantity level of inventory on hand that triggers a new purchase order. The reorder point is simplest to compute when both demand and the purchase-order lead time are known with certainty:

$Reorder point = \begin{matrix} Number of units sold \\ per time period \end{matrix} \times \begin{matrix} Purchasee - order \\ lead time \end{matrix}$

Suppose the purchase-order lead time for UX1s is 2 weeks:

Economic order quantity	1,000 units
Number of units sold per week	250 units per week $inline style left parenthesis text 13,000?units? end text back space text ?? end text divided by ? back space 52 thin space text weeks end text right parenthesis$
Purchase-order lead time	2 weeks

$Reorder point = 250 units per week \times 2 weeks = 500 units$

Watch Try It Video

Glare Shade will order 1,000 units of UX1s each time its inventory falls to 500 units.^* Exhibit 21-2 shows the behavior of the inventory level of UX1 units, assuming demand occurs uniformly during each week. If the purchase-order lead time is 2 weeks, a new order will be placed when the inventory level falls to 500 units, so the 1,000 units ordered will be received at the precise time that inventory reaches zero.

Exhibit shows graph depicting inventory level of UX1 sunglasses at Glare Shade.

X axis shows weeks ranging from 0 to 8 with increments of 1. Y axis shows inventory in Units marked as 0, 500 and 1000. On X axis distance between values 2 to 4 and 6 to 8 is marked as "Lead Time 2 weeks". Two lines are drawn with a steep downward slope, one between (0, 1000) on Y axis and (4,0) and other between (4, 1000) to (0, 8). Points labeled "Reorder Point" are marked on both lines. For first curve reorder point is (2, 500) and for second point is (6, 500).

The Economic-Order-Quantity Decision Model

Exhibit 21-1

Graphic Analysis of Ordering Costs and Carrying Costs for UX1 Sunglasses at Glare Shade

When to Order, Assuming Certainty

Watch Try It Video

Exhibit 21-2

Inventory Level of UX1 Sunglasses at Glare Shade^a