Revisited: Economic Order Quantity (EOQ)

The economic order quantity, or EOQ, is a fundamental mathematical formula in supply chain management. It was first developed by the engineer Ford Whitman Harris in 1913 soon after leaving the Westinghouse Electric and Manufacturing Company (now the media conglomerate ViacomCBS).

In essence, EOQ determines the optimal order quantity to purchase in order to minimize total costs. Consider a local grocery store ordering apples from a farmer to sell at its store. How much should be ordered? If the store orders just one apple, the first customer to buy it will surely cause the store to stockout and lose revenue on lost sales. Conversely, if the store orders too much, it will take up unnecessary shelf space that could otherwise be used to sell other produce. In time, the excess inventory will also spoil, thus causing further financial loss. EOQ helps calculate the ideal order quantity to minimize these costs.

Derivation

To understand the inputs and output of EOQ, one place to start is demonstrating how the formula is derived. In general, the total cost (T) for a specific good is the sum of the purchase cost, ordering cost, and holding cost, where:

• Purchase Cost is the unit product price (P) ⨉ annual demand quantity (Q).

• Ordering Cost is the fixed cost to order (K) ⨉ the number of orders per year, i.e. the annual demand divided by the order quantity (D/Q).

• Holding Cost is the unit product holding cost (H) ⨉ the average of the quantity on hand (Q/2).

Formulaically, this is represented as:

T = PD + K(D/Q) + H(Q/2)

A few distinctive assumptions apply here:

• Demand is uniform and deterministic for a period of one year.

• Supply is instantaneous, unlimited replenishment.

• Ordering Cost is fixed, regardless of order size.

Ordering costs are costs associated with ordering the inventory, such as shipping, handling, and packaging. Holding costs are costs associated with overhead expenses to carry inventory, such as warehouse space, insurance, depreciation, and even the opportunity cost of money held up in inventories

Figure 1 illustrates these costs curves. Recall from microeconomics that total cost increases as quantity increases, fixed costs decrease as quantity increases, and variable costs grow commensurately with quantity.

Visually, the EOQ formula is finding the minimum point of the total cost function (light green “TC” curve in Figure 1). From the formula, we determine this value by taking the derivate of total cost (T) with respect to the order quantity (Q), assuming all other variables are constant:

(dT/dQ) = –(KD/Q^2) + H/2

Then, setting the equation to 0 and solving for Q gives the optimal order quantity denoted as Q’:

0 = –(KD/Q^2) + H/2

Q’ = √((2DK)/H)

And that’s the EOQ formula! Note Q’ is a function of D, K, and H. In other words, the price (P) of the product does not matter. Only the demand, ordering cost, and holding cost are needed to determine EOQ.

Application

Referring back to the grocery store scenario, how many apples should the store order? Here are the inputs:

• Demand (D) for apples per year is 5,000.

• Product Price (P) per apple is $0.25.

• Ordering Cost (K) per order is $10.00.

• Holding Cost (H) per apple per year is 8% of product price.

But this is a real-world application, so let’s use Excel! There is no actual formula, but it can be recreated easily in Excel.

Caveats

Now that the basic EOQ formula has been derived and applied, it’s easier to see some of the real-world gaps in the equation:

• It assumes instantaneous replenishment without lead times.

• It assumes no negotiated purchase price discounts.

• It assumes uniform demand without seasonality or economic fluctuations.

• It assumes both ordering and holding costs are fixed.

There are more advanced formulas that incorporate these variables, and most EOQ calculations in business are done with complex enterprise resource planning systems. (Interestingly, adding lead time will not change the calculation of the economic order quantity if pipeline inventory is also incorporated in total cost.) Be cognizant of these assumptions and use the EOQ formula for simple calculations.

Summary

EOQ is the optimal order quantity that minimizes total ordering and holding costs. Not only does it help manage supply, but it also optimizes cash flow tied up in inventory. Despite its assumptions such as instantaneous replenishment and uniform demand, EOQ is a simple and effective formula all supply chain managers should understand.

References

Caplice, C; Ponce, E. (2020), SCM Key Concepts, MIT Center for Transportation and Logistics.

Hax, AC; Candea, D. (1984), Production and Operations Management, Prentice-Hall.