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Economic Order Quantity (EOQ) with Quantity Discounts

Prepared by:

Robbie HarmonBrigham Young University

November 28, 2005

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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

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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

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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.

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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?

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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

Order Size Holding Costs Ordering Costs

Too LARGE High Low

Too SMALL Low High

• EOQ helps you find the balance!!!

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EOQ

Order Quantity

Co

st

Holding Cost

Ordering Cost

Total Cost

EOQ is the quantity where Holding cost = Ordering cost

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Definition of EOQ ComponentsH = 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)

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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

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Solve for Q algebraically• (Q/2)H = (D/Q)S

• Q2 = 2DS/H

• Q = square root of (2DS/H) = EOQ

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When should we place an order for Q units?

• SS = safety stock

• Reorder point = ROP = d L + SS

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Introducing Quantity DiscountsWhat are quantity discounts?

Example:

Order Size 1 - 100 101 - 200 201 - 300

Price per unit $20 $18 $16

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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

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2 Steps1. 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.

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Limitations of this basic model

1. H and S are often estimated imprecisely

2. Ordering costs and demand rates vary throughout the year

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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”

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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)

• What is EOQ?

18EOQ = 316

EOQ

$0

$500

$1,000

$1,500

$2,000

$2,500

$3,000

100 150 200 250 300 350 400 450 500

Order Quantity

Co

st

Holding Cost

Ordering Cost

Total Cost

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Practice continued…• Now suppose the following quantity discounts are

available.

Order Size 1 - 200 201 - 350 351 - 500

Price per unit $20 $18 $16

• What amount should be purchased?

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Summary• Understanding EOQ and quantity discounts

can result in substantial savings!

• Review

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Review• What is EOQ?

• What 2 steps should be taken when considering quantity discounts?

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Reading ListBogner, Michael. “Quantity Discounts / Economic Order Quantity.”

http://www.dau.mil/pubs/pm/pmpdf02/Sept_Oct/BOGSE-0C2.pdf

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. 259 -267)

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)

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Reading ListSchreibfeder, Jon. “Effective Replenishment Parameters.” Microsoft. Microsoft

Business Solutions. http://download.microsoft.com/download/d/6/9/d69de816-aecb-4869-a920-2b5afccd7589/eimwp4_replenish.pdf

Toomey, John W. Inventory Management: Principles, Concepts, and Techniques. Norwell, MA: Kluwer Academic Publishers, 2000 (pp. 61-72)