# Economic Order Quantity (EOQ) with Quantity Discounts

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Economic Order Quantity (EOQ) with Quantity Discounts
Prepared by: Robbie Harmon Brigham Young University November 28, 2005

Outline What is EOQ? When do I use it? Definition of EOQ components
How does it work? Introducing Quantity Discounts Are there any limitations? Real World Example Practice Summary

What is EOQ? EOQ = mathematical device for arriving at the purchase quantity of an item that will minimize the cost equation below total cost = holding costs + ordering costs

So…What does that mean? Basically, EOQ helps you identify the most economical way to replenish your inventory by showing you the best order quantity.

When do I use it? Suppose you are responsible for ordering inventory. You have the following information. It costs \$5 to hold one widget in inventory for a year It costs \$100 to place an order for widgets, regardless of size Customers demand 2,500 widgets every year (Sales are distributed evenly throughout the year) How large should your orders be to minimize total cost?

How large should your orders be?
If your orders are too large, you’ll have excess inventory and high holding costs If your orders are too small, you will have to place more orders to meet demand, leading to high ordering costs You could, for example, order all 1,500 widgets at once but holding costs would be too high. Another possibility would be to place orders of 5 units at a time. That would lead to extremely high ordering costs. Order Size Holding Costs Ordering Costs Too LARGE High Low Too SMALL EOQ helps you find the balance!!!

EOQ is the quantity where Holding cost = Ordering cost
As illustrated by this graph, the lowest total cost is achieved by finding a balance between holding costs and ordering costs. In other words, finding the order quantity where holding costs = ordering costs yields the lowest total cost in this simplified model. EOQ is the quantity where Holding cost = Ordering cost

Definition of EOQ Components
H = annual holding cost for one unit of inventory S = cost of placing an order, regardless of size P = price per unit d = demand per period D = annual demand L = lead time Q = Order quantity (this is what we are solving for) Holding costs include: Cost of storage space Cost of potential obsolescence Opportunity cost of tying up the organizations funds in inventory (Bozarth, pg. 421) At this point, we’re assuming P and S are fixed, and L, d, and D are constant.

How does it work? Total annual holding cost = (Q/2)H
Total annual ordering cost = (D/Q)S EOQ: Set (Q/2)H = (D/Q)S and solve for Q H = annual holding cost for one unit of inventory S = cost of placing an order, regardless of size P = price per unit d = demand per period D = annual demand L = lead time Q = Order quantity (this is what we are solving for) We use Q/2 because inventory levels go from Q to zero over and over again. Thus, the average inventory level is Q/2. Multiply the average inventory level (Q/2) by the holding cost for one unit of inventory (H) to arrive at the total annual holding cost. We find the number of orders each year by dividing annual demand (D) by order quantity (Q). Multiply this quotient (D/Q) by the cost of placing a single order (S) to arrive at the total annual ordering cost.

Solve for Q algebraically
(Q/2)H = (D/Q)S Q2 = 2DS/H Q = square root of (2DS/H) = EOQ This tells us how much to order but not when to order. Next we’ll turn our attention to reorder point and safety stock calculations.

When should we place an order for Q units?
SS = safety stock Reorder point = ROP = d L + SS H = annual holding cost for one unit of inventory S = cost of placing an order, regardless of size P = price per unit d = demand per period D = annual demand L = lead time Q = Order quantity (this is what we are solving for) We’re still assuming P and S are fixed, and L, d, and D are constant. We need to order more inventory when inventory levels are at a point where they can meet demand while waiting for the new order. In other words, we need to have enough inventory on hand to meet demand during lead time. Safety stock is an extra amount beyond that needed to meet average demand during lead time. This guards against stock-outs and raises the reorder point. The calculation of Safety stock varies from company to company. We will not cover the details of its calculation here. Some companies set an arbitrary standard such as setting reorder point equal to 125% of expected demand. Safety stock should be determined by Variability of demand Variability of lead time Average length of lead time Desired service level When inventory reaches ROP it is time to order Q units. Average d times average L plus safety stock = ROP SS can

Introducing Quantity Discounts
What are quantity discounts? Example: At this point we assume that prices can vary. This complicates our EOQ model but only slightly. Order Size Price per unit \$20 \$18 \$16

EOQ with Quantity Discounts
Minimize the following equation: Total cost = holding costs + ordering costs + item costs (Total cost = (Q/2)H + (D/Q)S + DP) This is done in 2 steps H = annual holding cost for one unit of inventory S = cost of placing an order, regardless of size P = price per unit d = demand per period D = annual demand L = lead time Q = Order quantity (this is what we are solving for)

2 Steps Calculate EOQ. If this amount can be purchased at the lowest price, you have found the quantity that minimizes the equation. If not, proceed to step 2. Compare total cost at the EOQ quantity with total costs at each price break above the EOQ. In step 2 there is no need to look at those quantities below the EOQ, as these would result in higher holding and ordering costs, as well as higher item costs.

Limitations of this basic model
H and S are often estimated imprecisely Ordering costs and demand rates vary throughout the year Though estimates of H and S may be imprecise, this is not a great concern because these costs are relatively flat around EOQ so order quantities can be slightly off and still yield a fairly accurate answer. 2. While this is an issue, EOQ still provides a good starting point for understanding the impact of order quantity on costs

Real World example 1974 Report to Congress by the Comptroller General of the U.S. “Proper Use of the Economic Order Quantity Principle Can Lead to More Savings” In this report, the Comptroller General outlined the principles of EOQ and showed how various departments of the government could benefit from implementing EOQ.

Practice Suppose you are responsible for ordering inventory. You have the following information. It costs \$5 to hold one widget in inventory for a year It costs \$100 to place an order for widgets, regardless of size Customers demand 2,500 widgets every year (Sales are distributed evenly throughout the year) Remember… EOQ = square root of (2DS/H) Solution: = square root of [(2 x 2,500 x 100) / 5] = square root of (100,000) = E0Q = 316 What is EOQ?

EOQ = 316

Practice continued… Now suppose the following quantity discounts are available. Step 1 = compute EOQ Already found EOQ = 316 Can 316 be ordered at the lowest purchase price? No. Proceed to step 2 Step 2 = compare total cost at EOQ and total cost at price steps above EOQ Total cost = (Q/2)H + (D/Q)S + DP Total cost at EOQ = [(316/2) x 5] + [(2,500/316) x 100] + (2,500 x 18) Total cost at EOQ = ,000 = \$46,581 Total cost at 351 units = [(351/2) x 5] + [(2,500/351) x 100] + (2,500 x 16) Total cost at 351 units = ,000 = \$41,590 In this case we should purchase more than EOQ to take advantage of the quantity discount. Order Size Price per unit \$20 \$18 \$16 What amount should be purchased?

Summary Understanding EOQ and quantity discounts can result in substantial savings! Review

Review What is EOQ? What 2 steps should be taken when considering quantity discounts? What is EOQ? The quantity at which holding costs and ordering costs are equal. (Q/2)H = (D/Q)S What 2 steps should be taken when considering quantity discounts? Calculate EOQ. If this amount can be purchased at the lowest price, you have found the quantity that minimizes the equation. If not, proceed to step 2. 2. Compare total cost at the EOQ quantity with total costs at each price break above the EOQ.

Reading List Bogner, Michael. “Quantity Discounts / Economic Order Quantity.” Bozarth, Cecil C., & Handfield, Robert B. Introduction to Operations and Supply Chain Management. Upper Saddle River, NJ: Pearson Prentice Hall, 2005 Bragg, Steven M. Inventory Best Practices. Hoboken, NJ: John Wiley & Sons, Inc., 2004 Ozcan, Yasar A. Quantitative Methods in Health Care Management. San Francisco, CA: Jossey-Bass, 2005 (pp ) Report to the Congress by the Comptroller General of the United States. “Proper Use of the Economic Order Quantity Principle Can Lead to More Savings”: United States General Accounting Office, 1974 (pp. 1-10)

Reading List Schreibfeder, Jon. “Effective Replenishment Parameters.” Microsoft. Microsoft Business Solutions. Toomey, John W. Inventory Management: Principles, Concepts, and Techniques. Norwell, MA: Kluwer Academic Publishers, 2000 (pp )