Managing Facilitating Goods

Factory Wholesaler Distributor Retailer Customer

Replenishment order

Replenishment order

Replenishment order

Customer order

Production Delay

WholesalerInventory

Shipping Delay

Shipping Delay

DistributorInventory

RetailerInventory

Item Withdrawn

Learning Objectives

• Discuss the role of information technology in managing inventories.

• Describe the functions and costs of an inventory system.• Determine the order quantity.• Determine the reorder point and safety stock for inventory

systems with uncertain demand.• Design a continuous or periodic review inventory-control

system.• Conduct an ABC analysis of inventory items.• Determine the order quantity for the single-period inventory

case.• Describe the rationale behind the retail discounting model.

Role of Inventory in Services

• Decoupling inventories

• Seasonal inventories

• Speculative inventories

• Cyclical inventories

• In-transit inventories

• Safety stocks

Considerations in Inventory Systems

• Type of customer demand

• Planning time horizon

• Replenishment lead time

• Constraints and relevant costs

Relevant Inventory Costs

• Ordering costs

• Receiving and inspections costs

• Holding or carrying costs

• Shortage costs

Inventory Management Questions• What should be the order quantity

(Q)?

• When should an order be placed, called a reorder point (ROP)?

• How much safety stock (SS) should be maintained?

Inventory Models• Economic Order Quantity (EOQ)• Special Inventory Models

With Quantity DiscountsPlanned Shortages

• Demand Uncertainty - Safety Stocks• Inventory Control Systems

Continuous-Review (Q,r)Periodic-Review (order-up-to)

• Single Period Inventory Model

Inventory Levels For EOQ Model

0

Un

its o

n H

an

d

Q

Q

D

Time

Annual Costs For EOQ Model

0

100

200

300

400

500

600

700

800

9000 20 40 60 80 100

120

140

Order Quantity, Q

An

nu

al C

ost

, $

Holding CostOrdering CostTotal Cost

EOQ Formula

• NotationD = demand in units per yearH = holding cost in dollars/unit/yearS = cost of placing an order in dollarsQ = order quantity in units

• Total Annual Cost for Purchase Lots

• EOQ TCp S D Q H Q ( / ) ( / )2

EOQDS

H

2

Annual Costs for Quantity Discount Model

0 100 200 300 400 500 600 700

22,000

21000

20000

2000

1000

C = $20.00 C = $19.50 C = $18.75

Order quantity, Q

An

nua

l Co

st, $

Inventory Levels For Planned Shortages Model

Q

Q-K

0

-KT1 T2

TIME

T

Formulas for Special Models

• Quantity Discount Total Cost Model

• Model with Planned Shortages

TC CD S D Q I CQqd ( / ) ( / )2

TC SD

QH

Q K

QB

K

Qb

( )2 2

2 2

QDS

H

H B

B*

2

K QH

H B* *

Values for Q* and K* as AFunction of Backorder CostB Q* K* Inventory Levels

B

0 B

B 0

2DS

H

2DS

H

H B

B

undefined

QH

H B*

Q*

00

0

0

Demand During Lead Time Example

++ + =

u=3

15.

u=3 u=3 u=3

15. 15.

L 3

dL

12 ROP

s s

Four Days Lead Time Demand During Lead time

15.

Safety Stock (SS)

• Demand During Lead Time (LT) has Normal Distribution with - -

• SS with r% service level

• Reorder Point

Mean d LTL( ) ( )Std Dev LTL. .( )

SS z LTr

ROP SS dL

Continuous Review System (Q,r)

Average lead time usage, dL

Reorder point, ROP

Safety stock, SS

Inventory on hand

Ord

er q

uant

ity, E

OQ

EOQ

EOQ

d1 d2

d3

Amount used during first lead time

First leadtime, LT1

Order 1 placed

LT2 LT3

Order 2 placed Order 3 placed

Shipment 1 received Shipment 2 received Shipment 3 received

Time

Periodic Review System(order-up-to)

RP RP RP

Review period

First order quantity, Q1

d1

Q2Q3

d2

d3

Target inventory level, TIL

Amount used duringfirst lead time

Safety stock, SS First lead time, LT1 LT2 LT3

Order 1 placed Order 2 placed Order 3 placed

Shipment 1 received Shipment 2 received Shipment 3 received

Time

Inventory on Hand

Inventory Control Systems• Continuous Review System

• Periodic Review System

EOQDS

HROP SS LT

SS z LTr

2

RP EOQ

TIL SS RP LT

SS z RP LTr

/

( )

ABC Classification of Inventory Items

0102030405060708090

100

0 10 20 30 40 50 60 70 80 90 100

Percentage of inventory items (SKUs)

Perc

enta

ge o

f dol

lar v

olum

e

A B C

Inventory Items Listed in Descending Order of Dollar Volume

Monthly Percent of Unit cost Sales Dollar Dollar Percent of Inventory Item ($) (units) Volume ($) Volume SKUs Class

Computers 3000 50 150,000 74 20 AEntertainment center 2500 30 75,000

Television sets 400 60 24,000Refrigerators 1000 15 15,000 16 30 BMonitors 200 50 10,000

Stereos 150 60 9,000Cameras 200 40 8,000Software 50 100 5,000 10 50 CComputer disks 5 1000 5,000CDs 20 200 4,000

Totals 305,000 100 100

Single Period Inventory ModelNewsvendor Problem ExampleD = newspapers demanded

p(D) = probability of demand

Q = newspapers stocked

P = selling price of newspaper, $10

C = cost of newspaper, $4

S = salvage value of newspaper, $2

Cu = unit contribution: P-C = $6

Co = unit loss: C-S = $2

Single Period Inventory Model Expected Value Analysis

Stock Qp(D) D 6 7 8 9 10

.028 2 4 2 0 -2 -4

.055 3 12 10 8 6 4

.083 4 20 18 16 14 12

.111 5 28 26 24 22 20

.139 6 36 34 32 30 28

.167 7 36 42 40 38 36

.139 8 36 42 48 46 44

.111 9 36 42 48 54 52

.083 10 36 42 48 54 60

.055 11 36 42 48 54 60

.028 12 36 42 48 54 60

Expected Profit $31.54 $34.43 $35.77 $35.99 $35.33

Single Period Inventory Model Incremental Analysis

E (revenue on last sale) E (loss on last sale)

P ( revenue) (unit revenue) P (loss) (unit loss)

P D Q C P D Q Cu o( ) ( )

1 P D Q C P D Q Cu o( ) ( )

P D QC

C Cu

u o

( )

(Critical Fractile)

where: Cu = unit contribution from newspaper sale ( opportunity cost of underestimating demand) Co = unit loss from not selling newspaper (cost of overestimating demand) D = demand Q = newspaper stocked

Critical fractile for the newsvendor problem

0 2 4 6 8 10 12 14

Newspaper demand, Q

Pro

bab

ility

P(D<Q)(Co applies)

P(D>Q)(Cu applies)

0.722

Retail Discounting Model• S = current selling price• D = discount price• P = profit margin on cost (% markup as decimal)• Y = average number of years to sell entire stock of “dogs” at

current price (total years to clear stock divided by 2)• N = inventory turns (number of times stock turns in one year)

Loss per item = Gain from revenueS – D = D(PNY)

)1( PNY

SD

Topics for Discussion

• Discuss the functions of inventory for different organizations in the supply chain.

• How would one find values for inventory costs?• How can information technology create a competitive

advantage through inventory management?• How valid are the assumptions for the EOQ model?• How is a service level determined for inventory

items?• What inventory model would apply to service capacity

such as seats on an aircraft?

Interactive Exercise

The class engages in an estimation of the cost of a 12-ounce serving of Coke in various situations (e.g., supermarket, convenience store, fast-food restaurant, sit-down restaurant, and ballpark). What explains the differences?