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 Availability
1. 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
inventory
fr = 1 – 10/1000 = 1 – 0.01 = 0.99
Q = 1000
0
ESC = 10
inventory
0
time
fr = 1 – 970/1000 = 1 – 0.97 = 0.03
time
Q = 1000
ESC = 970
Expected Shortage per
Replenishment Cycle
• Expected shortage during the lead time

ESC 
 ( x  ROP ) f ( x)dx
where f(x) is pdf of DL
x  ROP
• If demand is normally distributed

 ss 
 ss 
   L f s 

ESC   ss 1  Fs 
  L 
L 

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
ESC
fr  1 
Q
 ss 
ESC   ss{1  F S  }
 L 
 ss 
  L f S  
 L 
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 policy
Lot size
Average demand during
lead time
Standard dev. of demand
during lead time
Expected shortage per
replenishment cycle
Q=
DL =
10,000
LD = 2*2,500 = 5,000
L =
SQRT(L)D =
SQRT(2)*500 = 707
ESC = -ss(1-Fs(ss/L))+Lfs(ss/L) =
-1000*(1-Fs(1,000/707) +
707fs(1,000/707) =
25.13
Product fill rate
fr =
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
Reorder point
Demand during lead time
0
Time
Lead time
L = SQRT(L2D + D2s2L)
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 inventory
Demand
Standard dev. of demand
Lead time
Demand during lead time
D=
D =
L=
DL =
Standard dev. of lead time sL =
Standard dev. of demand L =
during lead time
Safety inventory
ss =
2,500
500
7
LD = 5,000
7
SQRT(LD2 + D2sL2) =
SQRT(7*5002 + 25002*72) =
17,550
Fs-1(CSL)L =
Fs-1(0.90)*17,550 = 22,491
Summary
L: Standard deviation of
demand during lead time
sL: Standard deviation of
lead time
When lead time is constant
 L  L D
When lead time is uncertain
 L  L  D s
2
D
2 2
L
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
D

L
 LD

L
D
L
ROP  D L  ss
CSL  F ( ROP , D L , L )
ROP  F (CSL, D L , L )
1
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
ESC
fr  1 
Q
 ss 
ESC   ss{1  F S  }
 L 
 ss 
  L f S  
 L 
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?
Example Question
• You may use the table below to calculate the
safety inventory
Fs-1(0.9)
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)
None of the formulas can be
used to calculate the safety
inventory
1.28
314.08
340.62
564.08
590.62
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
Stddev of demand _ D
Lead time
L
Demand during lead time D_L
Stddev of demand during lead time  _L
Cycle service level CSL
Safety inventory
ss
Total safety inventory for 25 stores 25*ss
100
40
1
100
40
0.95
65.79
1644.9
Demand
Stddev of demand
Lead time
Demand during lead time
Stddev of demand during lead time
Cycle service level
Safety inventory
Safety inventory for 1 warehouse
ss = Fs-1(CSL)L
D
_ D
L
D_L
 _L
CSL
ss
1*ss
2500
200
1
2500
200
0.95
328.97
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 = 100
Cost c = 20
Ordering Too Much…
Cost c = 20
Salvage value s = 5
Cost of overstocking
Co = c - s
Versus Ordering Too Little…
Sell price p = 100
Cost 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
Demand
D_i
400
500
600
700
800
900
1000
1100
1200
1300
1400
1500
1600
1700
Prob
p_i
0.01
0.02
0.04
0.08
0.09
0.11
0.16
0.2
0.11
0.1
0.04
0.02
0.01
0.01
Cost c = $45
Price p = $100
Salvage value s = $5
What is the expected profit?
Expected demand = ∑Dipi = 1,026 parkas
Example: Parkas at L.L. Bean
Demand
D_i
400
500
600
700
800
900
1000
1100
1200
1300
1400
1500
1600
1700
Prob
p_i
0.01
0.02
0.04
0.08
0.09
0.11
0.16
0.2
0.11
0.1
0.04
0.02
0.01
0.01
Sold
Unsold
units
units
400
600
500
500
600
400
700
300
800
200
900
100
1000
0
1000
0
1000
0
1000
0
1000
0
1000
0
1000
0
1000
0
Profit
19000
25000
31000
37000
43000
49000
55000
55000
55000
55000
55000
55000
55000
55000
Expected profit = ∑profitipi = $49,900
Cost c = $45
Price p = $100
Salvage value s = $5
Example: Parkas at L.L. Bean
Demand
D_i
400
500
600
700
800
900
1000
1100
1200
1300
1400
1500
1600
1700
Prob
p_i
0.01
0.02
0.04
0.08
0.09
0.11
0.16
0.2
0.11
0.1
0.04
0.02
0.01
0.01
CSL
0.01
0.03
0.07
0.15
0.24
0.35
0.51
0.71
0.82
0.92
0.96
0.98
0.99
1
(1-CSL)
0.99
0.97
0.93
0.85
0.76
0.65
0.49
0.29
0.18
0.08
0.04
0.02
0.01
0
1100
1200
1300
1400
1500
1600
1700
Expected
Expected
Marg. benefit
Marg. cost
5500 x 0.49 = 2695 500 x 0.51 = 255
5500 x 0.29 = 1595 500 x 0.71 = 355
5500 x 0.18 = 990 500 x 0.82 = 410
5500 x 0.08 = 440 500 x 0.92 = 460
5500 x 0.04 = 220 500 x 0.96 = 480
5500 x 0.02 = 110 500 x 0.98 = 490
5500 x 0.01 = 55 500 x 0.99 = 495
(1 – CSL)(p – c)
CSL(c – s)
Expected
Marg. profit
2440
1240
580
-20
-260
-380
-440
What is the optimal order quantity?
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
=
Cu
Cu + Co
O* = F-1(CSL*, , ) = NORMINV(CSL*, , )
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 items
Average 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 =
0.88
NORMINV(CSL*, , ) =
468