Lecture 21

Supply Chain Management

Lecture 21

Outline

• Today– Finish Chapter 11

• Sections 1, 2, 3, 7, 8– Skipping 11.2 “Evaluating Safety Inventory Given Desired Fill rate”

– Start with Chapter 12• Sections 1, 2, 3

– Section 2 up to and including Example 12.2

• Friday– Homework 5 online

• Due Thursday April 8 before class

• Next week– Finish Chapter 12– Start with Chapter 14

Managing Inventory in Practice• India’s retail market

– Retail market (not inventory) projected to reach almost $308 billion by 2010

– Due to its infrastructure (many mom-and-pop stores and often poor distribution networks) lead times are long

ss = Fs-1(CSL)L

Managing Inventory in Practice• Department of Defense

– DOD reported (1995) that it had a secondary inventory (spare and repair parts, clothing, medical supplies, and other items) to support its operating forces valued at $69.6 billion

– About half of the inventory includes items that are not needed to be on hand to support DOD war reserve or current operating requirements

Safety Inventory

A new technology allows books to be printed in ten minutes. Borders has decided to purchase these machines for each store. They must decide which

books to carry in stock and which books to print on demand using this technology. Would you

recommend Borders to use the new technology for best-sellers or for other books?

Measuring Product Availability1. Cycle service level (CSL)

• Fraction of replenishment cycles that end with all customer demand met

• Probability of not having a stockout in a replenishment cycle

2. Product fill rate (fr)• Fraction of demand that is satisfied from product in inventory• Probability that product demand is supplied from available

inventory

3. Order fill rate• Fraction of orders that are filled from available inventory

Product Fill Rate

ESC = 10

inventory

inventory

time

time

0

0

Q = 1000

Q = 1000

fr = 1 – 10/1000 = 1 – 0.01 = 0.99

fr = 1 – 970/1000 = 1 – 0.97 = 0.03

ESC = 970

Expected Shortage per Replenishment Cycle• Expected shortage during the lead time

• If demand is normally distributed

L

ROPx

Df(x)dxxfROPxESC of pdf is where)()(

LsL

Ls

ssf

ssFssESC

1

Does ESC decrease or increase with ss?

Product Fill Rate

• fr: is the proportion of customer demand satisfied from stock.

Probability that product demand is supplied from inventory.

• ESC: is the expected shortage per replenishment cycle (is the demand not satisfied from inventory in stock per replenishment cycle)

• ss: is the safety inventory

• Q: is the order quantity

LSL

LS

ssf

ssFssESC

Q

ESCfr

}1{

1

Example 11-3: Evaluating fill rate given a replenishment policy• Recall that weekly demand for Palms at B&M is

normally distributed, with a mean of 2,500 and a standard deviation of 500. The replenishment lead time is two weeks. Assume that the demand is independent from one week to the next. Evaluate the fill rate resulting from the policy of ordering 10,000 Palms when there are 6,000 Palms in inventory.

Example 11-3: Evaluating fill rate given a replenishment policyLot size Q =

Average demand during lead time

DL =

Standard dev. of demand during lead time

L =

Expected shortage per replenishment cycle

ESC =

Product fill rate fr =

10,000

LD = 2*2,500 = 5,000

SQRT(L)D = SQRT(2)*500 = 707-ss(1-Fs(ss/L))+Lfs(ss/L) =-1000*(1-Fs(1,000/707) +707fs(1,000/707) =

25.13

1 – ESC/Q = 1 – 25.13/10,000 = 0.9975

Cycle Service Level versus Fill Rate

What happens to CSL and fr when the safety inventory (ss) increases?

What happens to CSL and fr when the lot size (Q) increases?

Lead Time Uncertainty

Why do some firms have zero tolerance for early/late deliveries?

Example 11-6: Impact of lead time uncertainty on safety inventory

Inventory

Time0

Reorder point

Lead time

Demand during lead time

L = SQRT(L2D + D2s2

L)

Example 11-6: Impact of lead time uncertainty on safety inventory• Daily demand at Dell is normally distributed, with

a mean of 2,500 and a standard deviation of 500. A key component in PC assembly is the hard drive. The hard drive supplier takes an average of L = 7 days to replenish inventory at Dell. Dell is targeting a CSL of 90 percent for its hard drive inventory. Evaluate the safety inventory of hard drives that Dell must carry if the standard deviation of the lead time is 7 days.

Example 11-6: Impact of lead time uncertainty on safety inventoryDemand D =

Standard dev. of demand D =

Lead time L =

Demand during lead time DL =

Standard dev. of lead time

sL =

Standard dev. of demand during lead time

L =

Safety inventory ss =

2,500

500

LD = 5,000

SQRT(LD2 + D2sL

2) = SQRT(7*5002 + 25002*72) =17,550

Fs-1(CSL)L =

Fs-1(0.90)*17,550 = 22,491

7

7

Summary

222LDL sDL

L: Standard deviation of demand during lead time

sL: Standard deviation of lead time

DL L When lead time is uncertain

When lead time is constant

Summary

L: Lead time for replenishment

D: Average demand per unit time

D:Standard deviation of demand per period

DL: Average demand during lead time

L: Standard deviation of demand during lead time

CSL: Cycle service level

ss: Safety inventory

ROP: Reorder point

),,(

),,(1

LL

LL

L

DL

L

DD

D

D

CSLFROP

ROPFCSL

ssROP

L

LD

Average Inventory = Q/2 + ss

Summary

• fr is the product fill rate (fraction of demand satisfied from inventory)

• ESC is the expected shortage per replenishment cycle (the demand not satisfied from inventory per replenishment cycle)

• ss is the safety inventory

• Q is the order quantity

LSL

LS

ssf

ssFssESC

Q

ESCfr

}1{

1

Example Question

• Weekly demand for canned fruit at a grocery store is normally distributed, with a mean of 250 and a standard deviation of 50. The lead time is two weeks. Assuming a continuous review replenishment policy, how much safety inventory should the store carry to achieve a CSL of 90 percent?

1.28314.08340.62564.08590.62

None of the formulas can be used to calculate the safety inventory

Example Question

• You may use the table below to calculate the safety inventory

F-1(0.9, 250, 50)F-1(0.9, 250, 70.71)F-1(0.9, 500, 50)F-1(0.9, 500, 70.71)

Fs-1(0.9)

Safety Inventory

Why is Amazon.com able to provide a large variety of books and music with less safety inventory than a

bookstore chain selling through retail stores?

Borders versus Amazon

~500 Borders stores versus ~20 Amazon warehouses

Demand D 100Stddev of demand _ D 40

Lead time L 1Demand during lead time D_L 100

Stddev of demand during lead time _L 40Cycle service level CSL 0.95

Safety inventory ssTotal safety inventory for 25 stores 25*ss




Safety inventory ssSafety inventory for 1 warehouse 1*ss

ss = Fs-1(CSL)L




Safety inventory ss 65.79Total safety inventory for 25 stores 25*ss 1644.9




Safety inventory ss 328.97Safety inventory for 1 warehouse 1*ss 328.97

Amazon versus Borders

“Company-wide, Borders has knocked eight days off of its days inventory outstanding through

improvements in its supply chain. Nevertheless, inventory stuck around 176 days in 1999, turning just

over twice a year. That's not very often. Barnes & Noble turned its inventory 2.5 times last year, and

Amazon managed nine turns. If Borders could turn its inventory as often as Barnes & Noble, it would free up

an additional $400 million for use during the year.”

Soure: Brian Lund (TMF Tardior), May 19, 2000

Safety Inventory

In the 1980s, paint was sold by color and size in paint retail stores. Today paint is mixed at the paint store according to the color desired. What impact did this

change had on safety inventories in the supply chain?

Importance of the Level of Product Availability• Product availability (also known as customer

service level) is measured by – CSL (Cycle service level)– fr (Product fill rate)

• Product availability affects supply chain responsiveness and costs– High levels of product availability increased

responsiveness and higher revenues– High levels of product availability increased

inventory levels and higher costs

The Newsboy/Newsvendor Problem

The Newsboy/Newsvendor Problem• One time decision under uncertainty

– Demand is uncertain– Plan inventory for a single cycle

• Trade-off– Ordering too much

• (waste, salvage value < cost)

– Ordering too little• (excess demand is lost)

• Examples– Restaurants– Fashion– High tech

The Christmas Tree Problem

Sell price p = 100Cost c = 20

Ordering Too Much…

Cost c = 20Salvage value s = 5

Cost of overstocking Co = c - s

Versus Ordering Too Little…

Sell price p = 100Cost c = 20

Cost of understocking Cu = p - c

Factors Affecting the Optimal Level of Product Availability

• Cost of overstocking (Co = c – s)– The loss incurred by a firm for each unsold unit at the

end of the selling season

• Cost of understocking (Cu = p – c)– The margin lost by a firm for each lost sale because

there is no inventory on hand• Includes the margin lost from current as well as

future sales if the customer does not return

Product Availability

• Cost of overstocking– Liz Claiborne experiences “unexpected earnings

decline as a consequence of “higher-than-expected excess inventories”

• The Wall Street Journal, July 19, 1993

– “On Tuesday, the network-equipment giant Cisco provided the grisly details behind its astonishing $2.25 billion inventory write-off in the third quarter”

• News.com, May 9, 2001

• Cost of understocking– IBM struggles with shortages in ThinkPad line due to

ineffective inventory management• The Wall Street Journal, August 24, 1994

Example: Parkas at L.L. Bean

Expected demand = ∑Dipi = 1,026 parkas

Demand ProbD_i p_i

400 0.01500 0.02600 0.04700 0.08800 0.09900 0.11

1000 0.161100 0.21200 0.111300 0.11400 0.041500 0.021600 0.011700 0.01

Cost c = $45

Price p = $100

Salvage value s = $5

What is the expected profit?


Expected profit = ∑profitipi = $49,900

Demand ProbD_i p_i

400 0.01500 0.02600 0.04700 0.08800 0.09900 0.11

1000 0.161100 0.21200 0.111300 0.11400 0.041500 0.021600 0.011700 0.01

Sold Unsold Profitunits units

400 600 19000500 500 25000600 400 31000700 300 37000800 200 43000900 100 49000

1000 0 550001000 0 550001000 0 550001000 0 550001000 0 550001000 0 550001000 0 550001000 0 55000

Cost c = $45

Price p = $100

Salvage value s = $5


Demand ProbD_i p_i

400 0.01500 0.02600 0.04700 0.08800 0.09900 0.11

1000 0.161100 0.21200 0.111300 0.11400 0.041500 0.021600 0.011700 0.01

CSL (1-CSL)

0.01 0.990.03 0.970.07 0.930.15 0.850.24 0.760.35 0.650.51 0.490.71 0.290.82 0.180.92 0.080.96 0.040.98 0.020.99 0.01

1 0

Expected Expected Expected Marg. benefit Marg. cost Marg. profit

1100 5500 x 0.49 = 2695 500 x 0.51 = 255 2440


1100 5500 x 0.49 = 2695 500 x 0.51 = 255 24401200 5500 x 0.29 = 1595 500 x 0.71 = 355 1240


1100 5500 x 0.49 = 2695 500 x 0.51 = 255 24401200 5500 x 0.29 = 1595 500 x 0.71 = 355 12401300 5500 x 0.18 = 990 500 x 0.82 = 410 5801400 5500 x 0.08 = 440 500 x 0.92 = 460 -201500 5500 x 0.04 = 220 500 x 0.96 = 480 -2601600 5500 x 0.02 = 110 500 x 0.98 = 490 -3801700 5500 x 0.01 = 55 500 x 0.99 = 495 -440

What is the optimal order quantity?

(1 – CSL)(p – c) CSL(c – s)

Optimal Level of Product Availability• Expected marginal contribution of raising the

order size from O* to O*+1(1 – CSL*)(p – c) – CSL*(c – s)

CSL* = Prob(Demand O*) = =p – c

p – s

O* = F-1(CSL*, , ) = NORMINV(CSL*, , )

Cu

Cu + Co

Example 12-1: Evaluating the optimal service level for seasonal items• The manager at Sportmart, a sporting goods store, has

to decide on the number of skis to purchase for the winter season. Based on past demand data and weather forecasts for the year, management has forecast demand to be normally distributed, with a mean 350 and a standard deviation of 100. Each pair of skis costs $100 and retails for $250. Any unsold skis at the end of the season are disposed of for $85. Assume that it costs $5 to hold a pair of skis in inventory for the season. How many skis should the manager order to maximize expected profits?

Example 12-1: Evaluating the optimal service level for seasonal itemsAverage demand (mean) =Standard deviation of demand (stdev)

=

Material cost c =Price p =Salvage value s =Cost of understocking Cu =

Cost of overstocking Co =

Optimal cycle service level CSL* =

Optimal order size O* =

350

100

$100

$250

85 – 5 = $80

p – c = 250 – 100 = $150

c – s = 100 – 80 = $20

Cu/(Cu + Co) = 150/170 = 0.88NORMINV(CSL*, , ) = 468

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Lecture 21

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