2 Inventory Management

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Inventory Management - Victor Araman

Inventory and (Yield) Management

Best Buy’s Lesson

In the 1990s, Eric Morley, Best Buy’s director of transportation,

remembers $15 million worth of personal computers were on the

way to stores in time for the holidays when chip maker Intel Corp.

unexpectedly announced it was going to introduce a new Pentium

processor, which wouldn’t be available until after the New Year. “We

were stuck,” Morley says. It became the Christmas without a PC. That’s

when we learned

that you don’t buy inventory just in case - you buy it just in

time”

Inventory Management – Victor Araman

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A First Look at Inventory Management

– Motivation

Continuous Replenishment Model

– Economic Order Quantity (EOQ)

– EOQ variants

Periodic Review Model

– News Vendor

– LL Bean (see other slides)

Supply Chain Inventory

Agenda



– Motivation



– EOQ variants


– News Vendor



Agenda

3

Inventory is the stock or store of an item or a resource used by

an organization.

Different types of inventory

Finished Goods (FGI) | Work in Process (WIP) | Raw Material

Examples

Cash in an ATM/bank

Half assembled engines in a car manufacturing plant

Phones in a retail store

Silicon in a semiconductor manufacturing plant

Seats in a plane

Advertising slots for a TV broadcaster

What Is Inventory?



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2-Mar-12 25-Feb-11 26-Feb-10

Cash And Cash

Equivalents1,199,000 1,103,000 1,826,000

Short Term

Investments- 22,000 90,000

Net Receivables 2,288,000 2,348,000 2,020,000

Inventory 5,731,000 5,897,000 5,486,000

Other Current

Assets1,079,000 1,103,000 1,144,000

10,297,000 10,473,000 10,566,000

140,000 328,000 324,000

3,471,000 3,823,000 4,070,000

1,335,000 2,454,000 2,452,000

359,000 336,000 438,000

- - -

403,000 435,000 452,000

- - -

16,005,000 17,849,000 18,302,000

Deferred Long Term Asset Charges

Total Assets

Long Term Investments

Property Plant and Equipment

Goodwill

Intangible Assets

Accumulated Amortization

Other Assets

Period Ending

Assets

Current Assets

Total Current Assets

Inventory 5,731,000 5,897,000 5,486,000

30-36% of

Total Assets

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Importance of Inventory

Inventories represent a major commitment of monetary resources

Inventories affect virtually all aspects of a company’s daily operations

Inventories represent a lethal “weapon”

Example: Dell, Wal-Mart, Zara

Importance of Inventory


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To meet anticipated customer demand

To protect against stock-outs

To take advantage of economic order cycles

To maintain independence of operations

To allow for smooth and flexible production operations

To hedge against inflation and price increases

To take advantage of quantity discounts

Why Do Companies Hold Inventory?


Inventory Sales Ratio


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Inventory Sales Ratio


Impact of Inventory on Valuation


Abnormal inventory growth vs. On year ahead earnings

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WSJ-09-09-04

Why Do Companies Hold Inventory?


Average aggregate inventory value:

– Average of the total value of all items held in inventory

Weeks of supply =

Inventory turns =

Some of these measures can be customized for specific

settings/industry

– Retail: sales per sqm of shelf space

– Restaurant: sales per available seat-hour

– Media: sales per slot per impression

weekper Sold Goods of Cost

valueinventory aggregate Average

valueinventory aggregate Average

weekper Sold Goods of Cost

Inventory Measures

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Common industry benchmark

Example (K-Mart: 2002)

– Inventory value = 4.825 B $

– Cost of Goods Sold = 26.258 B $/year

– Average time to turn a dollar (cost) to a dollar

= 4.825/26.258 year = 0.183 years = 67 days. (Little’s law)

• What is the financial significance of this time?

• Kmart improved this figure from 88 days in 1998.

– Kmart Inventory Turns in 2002 = (1/0.183) = 5.44 Turns/year

Example (Walmart: 2002)

– Inventory value = 22.75 B$, COGS = 171.56 B $

– Time to turn a dollar = 43 days

– Inventory Turns = 7.7

Inventory Turns


Consider a retailer whose inventory holding cost is 0.3 $/$/year (we will discuss the components of inventory costs later)

– That is, incur 30 cents to hold one unit of a good that costs 1 $ for 1 year

– Say, this retailer turns inventory 4 times per year

– So, on average, each unit stays in inventory for (1/4) of a year

– So, the cost that the retailer incurs just due to holding inventory is 30/4 = 7.5 cents per dollar

– In other words, when we buy a product for $100 at this retailer, $7.5 are paid towards the retailer’s inventory holding costs

Inventory Turns: Importance

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Think about profit margins for two Retailers (A and B) in the same market (selling the same products)

– If Retailer A turns 4 times per year, 7.5 cents is the inventory cost/$.

– If Retailer B turns 8 times per year, only 3.75 cents is the inventory cost per $ (by better Inventory management)

– All else being the same, the profit margin of B is 3.75 % higher than A!

– Typically, net profits in the retail industry can be as low as 2 % !

Inventory Turns: Importance


Compare:

Wal-Mart : 7.54 turns per year

K-Mart : 5.44 turns per year

Inventory Turns in the Retail Sector

20%

25%

30%

35%

40%

45%

0 5 10 15

Gro

ss M

arg

in

Inventory Turns

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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

ABC Analysis


0

20

40

60

80

100

0 50 100

% of Inventory Items

% Annual $ Usage

A

B C

Class % $ Vol % Items

A 80 15 B 15 30

C 5 55

Classifying Items as ABC

80-20 rule

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Ordering cost. (per order/transaction)

– cost incurred each time an order is placed with a supplier or production is ordered with its own shop

Holding cost. (per unit of inventory per unit time)

– cost associated with maintaining an item in inventory until it is used or sold

Stockout or shortage cost. (per unit of lost sale)

– occurs when the demand for an item exceeds its supply

Item cost. (per unit of inventory)

– Under constant demand: becomes relevant if a quantity discount is available

Inventory Decisions Driven by Cost



Housing costs

Material handling costs

Labor cost from extra handling

Investment costs

Pilferage, scrap, and obsolescence

6% (3 - 10%)

3% (1 - 3.5%) 3% (3 - 5%) 11% (6 - 24%)

3% (2 - 5%)

(Approximate Ranges)

Inventory Holding Costs

Category Cost as % of Inventory Value

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– Motivation



– EOQ variants


– News Vendor



Agenda


Economic Order Quantity - EOQ

Time

Inventory Level

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Single product or item

Demand rate known and constant

Item produced in lots, or purchased in orders

Each lot or order received in single delivery

Lead time known and constant

Ordering, or setup costs are constant

No backorders are allowed

No quantity discounts are allowed

Model Assumptions

Economic Order Quantity (EOQ)



Demand = D units per year

Ordering cost = S dollars per order placed

Holding cost = H dollars per unit per year

Order quantity = Q units

Data & Costs Formulation

– Holding Cost = H Q/2 per year

– Ordering Cost = S D/Q per year

– Total Cost = H Q/2 + S D/Q

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Order Quantity (Q)

Annual Cost

Order (Setup) Cost Curve

Q*

Optimal Order Quantity

H Q/2

S D/Q

EOQ Model: How Much to Order?


EOQ: When to Order?

Time

Inventory Level

Average

Inventory

Q* / 2

ROP Reorder Point

LT

Lead Time

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solution) (EOQ

)TC(

*

H

DSQ

QH

Q

DSQ

2

2

Solution of the EOQ Problem

Total Cost

Optimal Quantity


Expected Number of Orders

Expected Time between orders

Optimal Order Quantity

Reordering Point


D = Demand per year

S = Setup (order) cost per order

H = Holding (carrying) cost

d = Demand per day

LT = Lead time in days

EOQ Model Equations

= Q* H

2 × D × S = N D

Q*

ROP = d × LT Working Days per year = T

N

Working days / Year = d

D

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What if?

What if management is concerned by costs of

emissions?

What if there are discounts based on the order

quantity?

What if demand or/and lead times are NOT

deterministic?


EOQ Adjusted – Costs of Emissions


Cost of reducing emissions

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Out of Stock At Supermarkets



Safety Stock

Time

Inventory Level

ROP

P(Stockout)

dLT SS

Service

Level

Frequency

Avg dLT

Place order

Lead Time

ROP

Q

EOQ Adjusted – Demand Uncertainty

Avg dLT

Receive order

SS

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How much & when to order ?

Uncertain demand (and possibly uncertain lead time)

– Lead Time Demand: Demand during leadtime

– Leadtime demand follows normal distribution (forecasting)

– Continuous replenishment

What is a service level? What is the differ

– What is the difference between a service level and a fill rate?

How is the service level linked to the Safety Stock?

– Service level = 1 - Probability of stockout

– Higher service level means more safety stock

– More safety stock means higher ROP

– ROP = Avg of leadtime demand + safety stock

Service Level & Safety Stock


Leadtime demand is a normal distribution with an average

and standard deviation

Decide on your TARGET Service Level

– Likelihood that leadtime demand is smaller than ROP is a service

level

– Example: SL = 99% means that you leave a 1% chance of stock-out

during any cycle

Find the corresponding level of inventory: ROP – ROP = Avg dLT + Safety Stock

ROP Avg dLT

Normal

Using Excel the ROP is

ROP = NORMINV(SL, Avg dLT, stdev)

Leadtime Demand

Safety Stock Computation

Service

Level

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Demand during lead-time for one brand of TV is normally

distributed with a mean of 36 TVs and a standard deviation of 15

TVs. What safety stock should be carried for a 90% service level?

What is the appropriate reorder point?

Based on available information, the daily demand for CD-ROM

drives averages 10 units (normally distributed), with a standard

deviation of 1 drive. The lead-time is exactly 5 days. Management

wants a 97% service level. What safety stock should be carried?

What is the appropriate reorder point?

Probabilistic Model: Examples



– Motivation



– EOQ variants


– News Vendor



Agenda

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Inventory Management: The Newsvendor model


Single

Order

Demand

Uncertainty

Fixed

Price

Too

many?

Selling Newspapers…

Economic parameters

– buy at w = € 1

– sell at r = € 1.5

– salvage at s = € 0.05


Random demand

– a distribution is available

(with an average 300 and a

standard deviation 100 units)

The too much/too little problem

Order too much and there is a loss due to unsold newspapers Order too little and you lose potential sales (and profits)

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Market Uncertainty & Ex-Ante Bet

There are consequences of getting this bet wrong

The expected profit maximization balances the “too

much too little” costs Inventory Management – Victor Araman

Brief Comparative Analysis


Long lifecycle products with stationary demand

No demand uncertainty

Tradeoff between setup and

holding costs, driven by the

frequency of ordering

Short lifecycle products – One shot

items

– can be stocked only once at the beginning of the selling season)

Considerable demand uncertainty

Tradeoff between

– costs of excess leftover inventory (overstocking, holding or disposal costs)

– excess demand (stockout costs)

Examples

– fashion clothing | toys | computer games | music albums | books, consumer electronics

EOQ The “Newsvendor” model

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Overage cost per unit Co is the (opportunity) cost of one unit of

excess inventory (“over” ordering)

– What if you had ordered one fewer unit?

– Overage cost = Cost – Salvage value = w – s

• What if additional cost is required to dispose of a leftover inventory?

– Co = € 0.95

Underage cost per unit Cu is the (opportunity) cost of one unit of lost

sales (“under” ordering)

– What if you had ordered one additional unit?

– Underage cost = Price – Cost = r – w

• what if a goodwill cost or penalty cost is incurred in addition to the

lost margin?

– Cu = € 0.50

The Newsvendor Concept

The Newsvendor Order Quantity Q

We define the critical ratio

CR = 𝐶𝑢

𝐶𝑢+𝐶𝑜

CR measures the balance of power between marginal costs of

shortage and leftover

– how worse or better is too little compared to too much


Demand

Probability

Distribution

Q Avg

Dmd

Critical Ratio

CR

Risk of

leftover Risk of

shortage

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News Boy – Victor F. Araman

Expected Profit

If Q < D : Revenues = r Q ;

If Q > D : Revenues = r D ; Salvage value = s (Q – D )

Profit = r min{D , Q} + s (Q – D )+ – w Q

Penalty cost = p = 0

r x Sales s x Leftover


Cost of Demand Uncertainty

Gr(Q) = (r - w) E(D - Q)+ + (w – S) E(Q - D)+

Profit = r min{D , Q} + s (Q – D )+ - w Q

Cu : under-ordering

cost or lost margin Expected Lost

Sales Co : over-ordering cost

or the overage cost Expected leftover

The mismatch Cost

x

x

Risk free profit Mismatch Cost

Expected Profit

Pr (Q) = (r – w) ED – Gr (Q)

Simple

manipulations

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0

10

20

30

40

50

60

70

80

0 800 1600 2400 3200 4000 4800 5600 6400

Ex

pecte

d g

ain

or

loss

.

Expected marginal benefit

of understocking

Expected marginal loss

of overstocking

Marginal Analysis: Balancing the Risks

Ordering one more unit


Q+1

Q X

X – Cu

X + Co

D > Q

D ≤ Q

(X – Cu) x Prob{D > Q }

+

(X + Co) x Prob{D ≤ Q }

As more units are ordered the average benefit from

ordering one unit decreases (it becomes more likely

to be left with inventory)

while the average loss of ordering one more unit

increases (it becomes less likely to be short in

inventory)

X – Cu Prob{D > Q } + Co Prob{D ≤ Q }

Q+1 is better than Q if

Co Prob{D ≤ Q } < Cu Prob{D > Q


Optimal Rule

Expected cost of ordering one extra unit = Expected cost of ordering one less unit

So, the optimal quantity solves for

C0 P(D ≤ Q) = Cu P(D > Q)

The quantity 𝐶𝑢

𝐶𝑢+𝐶𝑜 is known as the critical ratio.

In the newspaper example: critical ratio = 0.5/(0.95+0.5) = 0.345

Cost UnderageCost Overage

Cost UnderageQ}P{Demand

Newsvendor Cost Tradeoff

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Demand

Probability Distribution

Q Average

Demand

Newsvendor: Optimal Quantity to order

Critical Ratio

Risk of leftover

Risk of shortage

A Normal Distribution

If demand distribution is normal Use Excel


Q m =300

CR

0.345

s =100

Q* = Norminv(CR, m , s)

= Norminv(0.345,300,100)

= 260

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Financial Performance

If seller orders the newsvendor quantity Q*

– What should the seller expect in terms of profits ?

– What about averages sales?

– How many units will be discounted on average (salvage)?

– How much money is left on the table? (i.e. What is the fraction of

demand unmet?)

Once lost sales are evaluated the rest follow trivially

For normal distribution lost sales is known

– tables that provide the values of lost sales

– excel functions that give the exact value of lost sales

If demand is not normal. Harder to get (need simulation)


Average Lost Sales

Suppose demand can take one of these values

D belongs to {0,10, 20,…190, 200}

Suppose Q=120

What is the average lost sales?

– If D<=Q : No lost sales = 0

– If D = 130, lost sales = D - Q= 10

– If D = 140, lost sales = D - Q = 20

Average Lost Sales

= 10 x P(D=130) + 20 x P(D=140) + … + 80 x P(D=200)


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Average Lost Sales for Normal Dist.

Available information

– Demand (D) is Normal with avg: ED = 300 and stdev sD = 100

– Assuming the previous economic parameters

– then CR = 0.325 and Q = 260

Formula for Average Loss Sales

sD x L(z)

– s is the standard deviation (given)

– z = (Q – ED) / sD e.g. z = (260-300)/100 = – 0.4

L(z) = normdist(z, 0, 1, 0) – z x (1 - normdist(z, 0, 1, 1))

e.g. L(-0.4) = 0.63 and Average Lost Sales = 100 x 0.63 = 63


The Rest Indeed Follows

Sales + Lost Sales = Demand

Avg Sales = Avg Demand – Avg Lost Sales = 300 – 63 = 237units

Leftover Inventory + Sales = Q

Avg leftover inventory = Q – Avg Sales = 260 – 237=23units

Average Profit

Price x Avg Sales + Salvage x Avg Leftover Inv. – Cost * Q

= (Price – Cost) x Avg Sales – (Cost – Salvage) x Avg Leftover Inv.

= (1.5-1)*237-(1-0.06)*23 = $96.88 ~ $97


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Fare: New York - Chicago Wednesday to Friday

0

200

400

600

800

1000

1200

1400

US

$

August September October

0

200

400

600

800

1000

1200

1400

US

$

Fare: New York - Chicago Thursday to Saturday

August September October

Wednesday-Friday fares are 15-20% higher than Thursday-Saturday fares!

Yield Management in Action


“Yield Management: term used in many service industries

to describe techniques to allocate limited resources, such as

airplane seats or hotel rooms, among a variety of customers,

such as business or leisure travelers. – By adjusting this allocation a firm can optimize the total revenue or

"yield“ on the investment in capacity – Since these techniques are used by firms with extremely

perishable goods, or by firms with services that cannot be stored at

all, these concepts and tools are often called perishable asset

revenue management. – American Airlines credits yield management techniques for a

revenue increase of $500 million/year and Delta Airlines uses

similar systems to generate additional revenues of $300 million per

year.”

Yield Management

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“Marriott Hotels credits its yield management system for

additional revenues of $100 million per year, with

relatively small increases in capacity and costs

Broadcasting companies use yield management to

determine how much inventory (advertising slots) to sell

now to the "upfront market" and how much to reserve

and perhaps sell later at a higher price to the "scatter

market’’

The core logic is similar to the newsvendor model

Yield Management



– Motivation



– EOQ variants


– News Vendor



Agenda

31


A Simple Supply Chain

The manufacturer’s profit is given by

Pm = (w – m) Q*

where, m is the production unit cost incurred by the

manufacturer and Q* is the optimal quantity ordered by the

retailer

Manufacturer Retailer End

consumer

r, D w, Q m

Supply Chain Inventory Insights

Double marginalization effect

Postponement strategies

Risk sharing and supply chain maximum value


Refer to Newsvendor Problems

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Supply Contracts: Real Options

Retailer buys q “call” options at unit cost c

– Call option: right to buy one unit at the exercise price x

– Options bought ahead of season, but exercised after demand is observed

– Induce the retailer to buy more units at the expense of sharing the risk (demand uncertainty)

Retailer’s Expected profit

pr(q) = E[(R - x) min{D, q}] – c q

Comparing to Original Newsvendor

Max Expected Profit

r min{D , Q} + s (Q – D )+ - w Q

Solution given by Critical Ratio

– Cu = r – w | Co = w – s

– CR = 𝐶𝑢

𝐶𝑢+𝐶𝑜

– P(D ≤ Q) = CR

– Q*= norminv(CF, Avg D, sD)

Profit Manufacturer

(w – m) Q*


Max Profit

(r - x) min{D, q} – c q

Set

r → r – x | w → c | s → 0

By analogy, solution must be given

by Critical Ratio

– ku = r – x – c | ko = c

– CF = 𝑘𝑢

𝑘𝑢+𝑘𝑜

– P(D ≤ q) = CF

– q*= norminv(CF, Avg D, sD)

Profit Manufacturer

E (c - m)q + x min{D, q} + S (q - D)+

Standard Newsvendor Newsvendor with Options

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Retailer’s Expected Profit

Retailer’s Expected profit

pr(q) = E[(R - x) min{D, q}] – c q

Re-writing the profit

pr(q) = (R - x - c) ED – gr(q)

gr(q) = (R - x - c) E[(D - q)+] + c E[ (q - D)+] – ku = R - x - c : opportunity cost (too few options)

– k0 = c : opportunity cost (too many options)

gr(q*) = s (ku + k0) Normdist(Normsinv(ku /(ku + k0)),0,1,false)


Manufacturer’s Expected Profit

Expected profit

pm(q) = E[(c - M) q + x min{D, q} + S (q - D)+]

See notes for an Excel formulation of this profit under normal

demand!

Net revenues from

selling q options

Revenues from

exercised options Revenues from non-

exercised options

Leftover Sales

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Inventory is the result of the imbalance between Supply and Demand

It serves as a buffer to – Smooth seasonality

– Reduce risk/cost of stockouts

– Alleviate production scheduling

– Take advantage of economies of scale

Inventory is not free

EOQ. Simple and Commonly used model – Tradeoff between setup and holding costs

Newsvendor. Commonly used for One Shot Items – Critical Ratio measures the imbalance between too much and too

little


Key Learnings

2 Inventory Management

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