Inventory ManagementEOQ & Quantity

Discounts

Inventory ManagementEOQ & Quantity

DiscountsOperations Management

P. Kalyanasundaram

Purposes of InventoryPurposes of Inventory

Meet anticipated demand Decouple production & distribution

permits constant production quantities

Take advantage of quantity discounts Hedge against price increases Protect against shortages

Types of InventoryTypes of Inventory

Raw Materials Work in progress (WIP) Finished products Maintenance, Repairs & Operating

Supplies

Disadvantages of Inventory

Disadvantages of Inventory

Higher costs Item cost (cost of the item) Ordering (or setup) cost

cost of forms, clerk’s wages, EDI system Holding (or carrying) cost

Building lease, insurance, money tied up

Difficult to control Hides production problems

Inventory CostsInventory Costs

What costs do we experience because we carry inventory?

Inventory CostsInventory Costs

Costs associated with inventory: Cost of the products Cost of ordering Cost of holding Cost of disposal

Shrinkage CostsShrinkage Costs

How much is stolen? 2% for discount, dept. stores, hardware,

convenience, sporting goods 3% for toys & hobbies 1.5% for all else

Where does the missing stuff go? Employees: 44.5% Shoplifters: 32.7% Administrative / paperwork error: 17.5% Vendor fraud: 5.1%

Inventory Holding Costs

Inventory Holding Costs

Category % of Value

Housing (building) cost 6%

Material handling 3%

Labor cost 3%

Opportunity/investment 11%

Pilferage/scrap/obsolescence 3%

Total Holding Cost 26%

ABC Analysis

Divides on-hand inventory into 3 classes A class, B class, C class

Basis is usually annual $ volume $ volume = Annual demand x Unit cost

Policies based on ABC analysis Develop class A suppliers more Give tighter physical control of A items Forecast A items more carefully

Classifying Items as ABC

0

20

40

60

80

100

0 50 100 150

0

20

40

60

80

100

0 50 100 150

% of Inventory Items% of Inventory Items

% Annual $ Usage% Annual $ Usage

AA

BB CC

ABC Classification Solution

Stock # Vol. Cost $ Vol. % ABC

206 26,000 $ 36 $936,000

105 200 600 120,000

019 2,000 55 110,000

144 20,000 4 80,000

207 7,000 10 70,000

Total 1,316,000

ABC Classification Solution

Stock # Vol. Cost $ Vol. % ABC

206 26,000 $ 36 $936,000 71.1 A

105 200 600 120,000 9.1 A

019 2,000 55 110,000 8.4 B

144 20,000 4 80,000 6.1 B

207 7,000 10 70,000 5.3 C

Total 1,316,000 100.0

Inventory Models

Fixed order quantity models Economic order quantity Quantity discount

Fixed order period models

Economic Order Quantity

Economic Order Quantity

Assumptions Demand rate is known and constant No order lead time Shortages are not allowed Costs:

S - setup cost per order c - unit cost h - holding cost per unit time

EOQEOQ

Time

Inventory Level

Q*OptimalOrderQuantity

Decrease Due toConstant Demand

EOQEOQ

Time

Inventory Level

Q*OptimalOrderQuantity

InstantaneousReceipt of OptimalOrder Quantity

EOQEOQ

Time

Inventory Level

Q*

Lead Time

ReorderPoint(ROP)

EOQEOQ

Time

Inventory Level

Q*

Lead Time

ReorderPoint(ROP)

Average Inventory Q/2

Total CostsTotal Costs

Average Inventory = Q/2 Annual Holding costs = H * Q/2 # Orders per year = D / Q Annual Ordering Costs = S * D/Q Annual Total Costs = Holding + Ordering

Q

DS

QHQTC *

2*)(

How Much to Order?How Much to Order?

Annual Cost

Order Quantity

Holding Cost= H * Q/2

How Much to Order?How Much to Order?

Annual Cost

Order Quantity

Holding Cost= H * Q/2

Ordering Cost= S * D/Q

How Much to Order?How Much to Order?

Annual Cost

Order Quantity

Total Cost= Holding + Ordering

How Much to Order?How Much to Order?

Annual Cost

Order Quantity

Total Cost= Holding + Ordering

Optimal Q

Optimal QuantityOptimal Quantity

Q

DS

QH *

2* Total Costs =

2*

2 Q

DS

H

Take derivative with respect to Q =

Solve for Q:

22 Q

DSH

Set equal to zero0

H

DSQ

22 H

DSQ

2

Adding Lead TimeAdding Lead Time

Use same order size Order before inventory depleted ROP = Safety Stock + CR * LT

CR = Consumption Rate LT = Lead Time

H

DSQ

2

A Question:A Question:

If the EOQ is based on so many horrible assumptions that are never really true, why is it the most commonly used ordering policy?

Benefits of EOQ Benefits of EOQ

Even if conditions don’t hold perfectly, profits are close to optimal

Estimated parameters will not throw you off very far

Example: ABC Plumbing Supply Company

Example: ABC Plumbing Supply Company

ABC Plumbing Supply Company stocks thousands of plumbing items sold to regional plumbers, contractors and retailers. Mr. Henry, the firm’s general manager, wonders how much money could be saved annually if EOQ were used instead of the firms present rules of thumb. He instructs Mary, an inventory analyst to conduct an analysis of one material only (Material #3025, a brass valve) to see if significant savings might result from using EOQ. Mary develops the following estimates from accounting information:

Accounting InformationAccounting Information

D = 10000 valves per year Q = 400 valves per order (present

order quantity) H = $0.40 per valve per year S = $5.50 per order

SolutionSolution

Mary calculates the present total annual stocking costs:

TSC1 = Q/2 * H + D/Q*S

= 400/2 * 0.4 + 10000/400 *5.5

= 80 + 137.5

= $217.50

The EOQ is calculatedThe EOQ is calculated

EOQ = √2DS/H

= √2 (10000)(5.5)/0.4

= √275000

= 524.4 Valves

The total annual stocking costs if EOQ were employed is calculatedThe total annual stocking costs if EOQ were employed is calculated

TSC2 = Q/2 * H + D/Q*S

= (524.4/2) * 0.4 + (10000/524.4) *5.5

= 104.88 + 104.88

= $209.76

Estimated savings = TSC1 - TSC2

= 217.50-209.76

= $7.74

ConclusionConclusion

Mary concludes that if the annual savings on this one material were applied to the thousands of items in inventory, the savings from EOQ would be significant.

Quantity DiscountsQuantity Discounts

How does this all change if price changes depending on order size?

Explicitly consider price Compare the total annual material

costs (TMC) TMC = Total annual stocking costs +

Annual acquisition cost

= TSC + D(ac)

Compute EOQ with each of the sales prices. Note that H is usually a function of sales price or production cost. For example H may be defined as 20 percent of the sales price. Therefore EOQ will change as H and ac will change.

Determine which EOQ from step 1 above is feasible. In other words, is the computed EOQ in the quantity range of price?

The total annual material cost (TMC) is computed for the feasible EOQ and the quantity at any price break with lower sales prices.

The order quantity with the lower total annual material cost (TMC) is the economic order quantity.

Discount ExampleDiscount Example

D = 10,000 S = $20 H = 20% of ac

Price Quantity EOQc = 5.00 Q < 500 633

4.50 501-999 6663.90 Q >= 1000 716

Discount PricingDiscount Pricing

Total Cost

Order Size500 1,000

Price 1 Price 2 Price 3

X 633

X 666

X 716

Discount PricingDiscount Pricing

Total Cost

Order Size500 1,000

Price 1 Price 2 Price 3

X 633

X 666

X 716

Discount ExampleDiscount Example

Order 666 at a time:Hold 666/2 * 4.50 * 0.2= $299.70Order 10,000/666 * 20 = $300.00Mat’l 10,000*4.50 = $45,000.00 45,599.70

Order 1,000 at a time:Hold 1,000/2 * 3.90 * 0.2=$390.00Order 10,000/1,000 * 20 = $200.00Mat’l 10,000*3.90 = $39,000.00 39,590.00

Discount Model Discount Model

1.Compute EOQ for each price

2.Is EOQ ‘realizeable’? (is Q in range?)

3.Compute total cost for this quantity

4.Select quantity/price with lowest total cost.

When to Reorder with EOQ Ordering

Reorder Point - When the quantity on hand of an item drops to this amount, the item is reordered

Safety Stock - Stock that is held in excess of expected demand due to variable demand rate and/or lead time. (Safety Stock to be considered while ROP is decided)

Continuous (Q) Review System Example: A computer company has annual demand of 10,000. They want to determine EOQ for circuit boards which have an annual holding cost (H) of $6 per unit, and an ordering cost (S) of $75. They want to calculate TC and the reorder point (R) if the purchasing lead time is 5 days.

Continuous (Q) Review System Example: A computer company has annual demand of 10,000. They want to determine EOQ for circuit boards which have an annual holding cost (H) of $6 per unit, and an ordering cost (S) of $75. They want to calculate TC and the reorder point (R) if the purchasing lead time is 5 days.

EOQ (Q)

Reorder Point (R)

Total Inventory Cost (TC)

units 500$6

$75*10,000*2

H

2DSQ

units 200days 5*days 250

10,000Time Lead x Demand DailyR

$3000$1500$1500$62

500$75

500

10,000TC

Quantity Discount Example: Collin’s Sport store is considering going to a different hat supplier. The present supplier charges $10 each and requires minimum quantities of 490 hats. The annual demand is 12,000 hats, the ordering cost is $20, and the inventory carrying cost is 20% of the hat cost, a new supplier is offering hats at $9 in lots of 4000. Who should he buy from?

Quantity Discount Example: Collin’s Sport store is considering going to a different hat supplier. The present supplier charges $10 each and requires minimum quantities of 490 hats. The annual demand is 12,000 hats, the ordering cost is $20, and the inventory carrying cost is 20% of the hat cost, a new supplier is offering hats at $9 in lots of 4000. Who should he buy from?

EOQ at lowest price $9. Is it feasible?

Since the EOQ of 516 is not feasible, calculate the total cost (C) for each price to make the decision

4000 hats at $9 each saves $19,320 annually.

hats 516$1.80

20)2(12,000)(EOQ$9

$101,66012,000$9$1.802

4000$20

4000

12,000C

$120,98012,000$10$22

490$20

490

12,000C

$9

$10