Transcript Safety stock - Kellogg School of Management

Supply Chain Management
Managing the Supply Chain
Key to matching demand with supply
Cost and Benefits of inventory
Economies of Scale
Inventory management of a retailer: EOQ + ROP
Levers for improvement
Safety Stock
Hedging against uncertainty
Role of lead time
Improving Performance
Centralization & Pooling efficiencies
Postponement
Accurate Response
What is a Supply Chain?
The Procurement
or supply system
Raw Material
supply points Movement/
Transport
The Operating
System
Raw Material
Movement/
Storage
Transport
The Distribution System
Manufacturing
PLANT 1
PLANT 2
PLANT 3
Movement/
Transport
Finished Goods
Movement/
Storage
Transport
WAREHOUSE
WAREHOUSE
WAREHOUSE
A
B
C
MARKETS
What makes a “good” Supply Chain?
Corporate Finance
CISCO
SELECTRON
Current Assets
BS/ 2000
BS/2000
Cash &
Equivalents
4,234
1,475.5
Short-term investments
1,291
958.6
AR
2,299
2,146.3
Inventories
1,232
Deferred Income
2,054
260.5
Total Current Assets
11,110
8,628.2
%11
3,787.3
%44
Inventories represent about %34 of current assets for a typical
company in the US; %90 of working capital.
For each dollar of GNP in the trade and manufacturing sector,
about 40cents worth of inventory was held.
Average logistics cost = 21c/dollar = %10.5 of GNP.
Another example of financial flows at
the corporate level: Mobile Handsets

Data:
– End of quarter inventory position from 10Q statements
– Sales from income statement (smooth 9month avg.)

How can financial flows help explain the difference in performance during
the downturn?
Nokia ($M) Inventory
Sales per month (9 month avg)
Time (months)
Ericson ($M) Inventory
Sales per month (9 month avg)
Time (months)
Motorola ($M) Inventory
Sales per month (9 month avg)
Time (months)
1999
12/31/99
$ 1,786
$ 1,437
1.24
$ 3,016
$
486
6.20
$ 3,707
$ 1,088
3.41
3/31/00
$ 1,925
$ 1,437
1.34
$ 4,031
$
486
8.29
$ 4,688
$ 1,088
4.31
2000
6/30/00
9/30/00
$ 2,290 $ 2,062
$ 1,437 $ 1,437
1.59
1.43
$ 4,612 $ 4,984
$
486 $
486
9.48
10.25
$ 5,446 $ 5,557
$ 1,088 $ 1,088
5.01
5.11
12/31/00
$ 2,014
$ 1,437
1.40
$ 4,671
$
486
9.60
$ 5,242
$ 1,088
4.82
3/31/01
$ 1,985
$ 1,692
1.17
$ 4,667
$
245
19.02
$ 4,533
$
830
5.46
2001
6/30/01
$ 1,622
$ 1,692
0.96
$ 2,947
$
245
12.01
$ 3,842
$
830
4.63
9/30/01
$ 1,774
$ 1,692
1.05
$ 2,647
$
245
10.79
$ 3,250
$
830
3.91
4
Mobile Handsets: How explain the
difference in performance?
$6B
Inventory
Flow Time
20wks
18
$5B
16
Motorola
$4B
14
Ericson
12
Ericson
$3B
10
8
$2B
Motorola
6
$1B
Nokia
4
2
$-
Nokia
0
8/28/99 12/6/99 3/15/00 6/23/00 10/1/00 1/9/01 4/19/01 7/28/01 11/5/01 8/28/99 12/6/99 3/15/00 6/23/00 10/1/00 1/9/01 4/19/01 7/28/01 11/5/01
5
Solectron’s executive officer
compensation plan
• Base compensation
• Bonuses
– “The Bonus Plan provides for incentive compensation ...
based on certain worldwide, site and individual
performance measures. Worldwide and site performance
are measured based on targets w.r.t. profit before taxes,
inventory turns, days sales outstanding and return on assets.
The Compensation Committee believes that these factors
are indicative of overall corporate performance and
shareholder value.” [compensation committee report FY 96]
• Long Term Incentive Compensation
Supply Chain Metrics
Costs of not Matching
Supply and Demand
• Cost of overstocking
– liquidation
– obsolescence
– holding
• Cost of under-stocking
– lost sales and customer goodwill
– lost margin
Why hold Inventory?
• Economies of scale
– Fixed costs of ordering/manufacturing
– Quantity discounts
– Trade Promotions
Cycle/Batch stock
• Uncertainty
– Information Uncertainty
– Supply/demand uncertainty
• Seasonal Variability
• Strategic
– Flooding, availability
Safety stock
Seasonal stock
Strategic stock
Cost of Inventory
• Physical holding cost
(out-of-pocket)
• Financial holding cost
(opportunity cost)
• Low responsiveness
– to demand/market
changes
– to supply/quality changes
Holding cost (H)
Costs Associated with Batches
• Ordering costs (S)
– Changeover of production line (Set-up)
– Transportation (Delivery)
– Receiving
• Holding costs (H = r C)
– Physical holding cost
– Cost of capital (r)
– Cost of obsolescence
Palü Gear: Retail Inventory Management
Annual jacket revenues at a Palü Gear retail store are roughly
$1M. Palü jackets sell at an average retail price of $325, which
represents a mark-up of 30% above what Palü Gear paid its
manufacturer. Being a profit center, each store made its own
inventory decisions and was supplied directly from the
manufacturer by truck. For each order up to 5000 jackets, the
manufacturer charges a flat fee of $2,200 for delivery. To
exploit economies of scale, stores typically orders 1500 jackets
each time it places an order. (Palü’s cost of capital is
approximately 20%.) What order size would you recommend
for a Palü store in current supply network?
Palü Gear: Evaluation of current policy
of ordering 1500 units each time
1. What is average inventory I?
 I=
 Annual cost to hold one unit H =
 Annual cost to hold I =
2. How often do we order?
 Annual throughput R =
 # of orders per year =
 Annual order cost =
3. What is total cost?
 TC =
Can we do better ?
Find the most economical order quantity
Method 1 : Enumerate
Number of units Number of
per order/batch Batches per
Q
Year: R/Q
50
62
100
31
150
21
200
15
250
12
300
10
350
9
400
8
450
7
500
6
510
6
520
6
530
6
540
6
550
6
600
5
650
5
700
4
750
4
800
4
850
4
900
3
1000
3
Annual
Setup Cost
$ 135,385
$
67,692
$
45,128
$
33,846
$
27,077
$
22,564
$
19,341
$
16,923
$
15,043
$
13,538
$
13,273
$
13,018
$
12,772
$
12,536
$
12,308
$
11,282
$
10,414
$
9,670
$
9,026
$
8,462
$
7,964
$
7,521
$
6,769
Annual
Holding Cost
$
1,250
$
2,500
$
3,750
$
5,000
$
6,250
$
7,500
$
8,750
$
10,000
$
11,250
$
12,500
$
12,750
$
13,000
$
13,250
$
13,500
$
13,750
$
15,000
$
16,250
$
17,500
$
18,750
$
20,000
$
21,250
$
22,500
$
25,000
Annual
Total Cost
$ 136,635
$
70,192
$
48,878
$
38,846
$
33,327
$
30,064
$
28,091
$
26,923
$
26,293
$
26,038
$
26,023
$
26,018
$
26,022
$
26,036
$
26,058
$
26,282
$
26,664
$
27,170
$
27,776
$
28,462
$
29,214
$
30,021
$
31,769
$160,000
Setup Cost
$140,000
Holding Cost
$120,000
Total Cost
$100,000
$80,000
$60,000
$40,000
$20,000
$0
100 200 300 400 500 600 700 800 900 1000
Order (batch) size Q
Economies of Scale:
Inventory Build-Up Diagram
R: Annual demand rate
(units/yr),
Q: Number of jackets per
replenishment order
• Number of orders per
year = R/Q.
• Average number of
jackets in inventory =
Q/2 .
Inventory
T = Q/R
Q
R = Demand rate
T
time between
orders
Order placed,
Order placed,
arrives immediately
Lead time=0
arrives immediately
Lead time=0
Time t
Economic Order Quantity
EOQ
R
S
H
Q
:
Demand per year,
:
Setup or Order Cost ($/setup; $/order),
:
Marginal annual holding cost ($/per unit per year), H
:
Order quantity.
2SR
QEOQ 
H
C
r
:
Cost per unit ($/unit),
:
Cost of capital (% / yr),
=rC
Total annual
costs
2 SRH
H Q/2: Annual
holding cost
S R /Q:Annual
setup cost
QEOQ
Batch Size Q
What do we learn from the EOQ formula ?
How does the ordering policy change if …
The product is a success and the demand picks up,
now we are selling 4 times the original demand…
The interest rates double up, so does our unit
inventory holding costs …
After investing in IT, we manage to reduce our fixed
ordering cost by half …
Optimal Economies of Scale:
For a Palü Gear retailer
R = 3077 units/ year
r = 0.20/year
C = $ 250 / unit
S = $ 2,200 / order
Unit annual holding cost = H = $50/unit-yr
Optimal order quantity = EOQ = 520 units
Number of orders per year = R/Q = 3077/520 = 5.91
Time between orders = Q/R = 8.78 weeks
Annual order cost = (R/Q)S = $13,018/yr
Average inventory I = Q/2 = 260 units
Annual holding cost = (Q/2)H = $13,018/yr
Take-Aways I
Batching & Economies of Scale
• Increasing batch size of production (or purchase) increases
average inventories (and thus cycle times).
• Average inventory for a batch size of Q is Q/2.
• The optimal batch size trades off setup cost and holding cost.
• To reduce batch size, one has to reduce setup cost (time).
• Square-root relationship between Q and (R, S):
– If demand increases by a factor of 4, it is optimal to
increase batch size by a factor of 2 and produce (order)
twice as often.
– To reduce batch size by a factor of 2, setup cost has to be
reduced by a factor of 4.
Role of Supply Leadtime L:
Palü Gear cont.
• The lead time from when a Palü Gear retailer places an
order to when the order is received is two weeks. If
demand is stable as before, when should the retailer
place an order?
• I-Diagram:
ROP
Order 2wks
placed
ROP =
Order
arrives
Two Key Decisions of Inventory
Management
1.
How much to order ?
Answer: EOQ
2. When to place an order ?
Answer: ROP Reorder Point
represents the amount of inventory on hand when we place a new order .
Delivery Lead Time = 0 (Instantaneous Delivery) then ROP = …….
Delivery Lead Time > 0 then ROP = …….
ROP is driven by:
Delivery Lead Time
Demand Uncertainty
Customer Service Level
A Key to Matching Supply and
Demand
When would you rather place your bet?
A
A:
B:
C:
D:
B
A month before start of Derby
The Monday before start of Derby
The morning of start of Derby
The winner is an inch from the finish line
C D
Demand uncertainty and
forecasting
Year
• 1
• 2
• 3
• 4
• 5
• 6
Demand
323
258
303
304
284
285
Forecast Error
Demand uncertainty and forecasting
• Forecasts depend on
– historical data
– “market intelligence”
• Forecasts are usually (always?) wrong.
• A good forecast has at least 2 numbers (includes a
measure of forecast error, e.g., standard deviation).
• The forecast horizon must at least be as large as the
lead time. The longer the forecast horizon, the less
accurate the forecast.
• Aggregate forecasts tend to be more accurate.
Palü Gear:
Service levels & Inventory management
•
In reality, a Palü Gear store’s demand fluctuates from week to
week. In fact, weekly demand at each store had a standard
deviation of about 30 jackets  assume normally distributed.
Recall that average weekly demand was about 60 jackets; the
order lead time is 2 weeks; fixed order costs are $2,200/order
and it costs $50 to hold one jacket in inventory during one year.
• Questions:
1. If the retailer uses the ordering policy discussed before (ROP
=120), what will the probability of running out of stock in a
given cycle be?
2. The Palü retailer would like the stock-out probability to be
smaller. How can she accomplish this?
3. Specifically, how does it get the service level up to 95%?
Example: say we increase ROP to
140 (and keep order size at Q = 520)
1. On average, what is the stock level when the
replenishment arrives?
2. What is the probability that we run out of stock
before a delivery arrives?
3. How do we get that stock-out probability down to
5%?
Lead Time Demand
Consider the lead time between the placement of an
order and the arrival of the order. L=2 weeks.
Demand per week, R ~ (60, 30)
The mean demand during the lead time = 2*60 = 120
Standard deviation of demand during lead time
=
=
(30)2 + (30)2
30
2
D lead time ~ N (120, 30
2
)
IF ROP=120
Probability of stock out (i.e.,
Probability that the demand during
lead time is greater than 120 ) = ?
Lead time
demand distribution
120
Mean Demand
During Lead Time
SAFETY STOCK
Safety stock increase
with the service level.
Increase reorder point
above average demand
during lead time, lower
the probability of
stockout! BETTER
SERVICE LEVEL!
e.g., % 95
ROP
Less demand variability
Small variance / SD
%95
ROP
120
Safety stock increases
with the variance (SD) of lead time demand
Higher Variability
Larger SD/ variance
%95
120
ROP
Safety Stock
I s  z*R L
Safety stock increases (decreases) with an increase
(decrease) in:
• demand variability or forecast error,
• delivery lead time for the same level of service,
• delivery lead time variability for the same level of
service.
Palü Gear: Determining the required
Reorder Point for 95% service
DATA:
R = 60 jackets/ week
H = $50 / jacket, year
S = $ 2,200 / order
R = 30 jackets/ week
standard deviation of weekly demand
L = 2 weeks
QUESTION:
What should safety stock be to insure a desired cycle service level of 95%?
ANSWER:
1. Determine  lead time demand
= 42.42
2. Required # of standard deviations z*
= 1.64
3. Reorder Point = 120+1.64*42.42= 190 jackets
4.
Safety stock
Is = 1.64*42.42=70 jackets
Review of Probability
The standard normal distribution F(z)
• Transform X = N(m,) to
z = N(0,1)
z = (X - m) / .
F(z) = Prob ( N (0,1) < z)
F(z)
0
• Transform
z
back,
knowing z*:
X* = m + z*.
z
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.1
3.2
3.3
0.00
0.5000
0.5398
0.5793
0.6179
0.6554
0.6915
0.7257
0.7580
0.7881
0.8159
0.8413
0.8643
0.8849
0.9032
0.9192
0.9332
0.9452
0.9554
0.9641
0.9713
0.9772
0.9821
0.9861
0.9893
0.9918
0.9938
0.9953
0.9965
0.9974
0.9981
0.9987
0.9990
0.9993
0.9995
0.01
0.5040
0.5438
0.5832
0.6217
0.6591
0.6950
0.7291
0.7611
0.7910
0.8186
0.8438
0.8665
0.8869
0.9049
0.9207
0.9345
0.9463
0.9564
0.9649
0.9719
0.9778
0.9826
0.9864
0.9896
0.9920
0.9940
0.9955
0.9966
0.9975
0.9982
0.9987
0.9991
0.9993
0.9995
0.02
0.5080
0.5478
0.5871
0.6255
0.6628
0.6985
0.7324
0.7642
0.7939
0.8212
0.8461
0.8686
0.8888
0.9066
0.9222
0.9357
0.9474
0.9573
0.9656
0.9726
0.9783
0.9830
0.9868
0.9898
0.9922
0.9941
0.9956
0.9967
0.9976
0.9982
0.9987
0.9991
0.9994
0.9995
0.03
0.5120
0.5517
0.5910
0.6293
0.6664
0.7019
0.7357
0.7673
0.7967
0.8238
0.8485
0.8708
0.8907
0.9082
0.9236
0.9370
0.9484
0.9582
0.9664
0.9732
0.9788
0.9834
0.9871
0.9901
0.9925
0.9943
0.9957
0.9968
0.9977
0.9983
0.9988
0.9991
0.9994
0.9996
0.04
0.5160
0.5557
0.5948
0.6331
0.6700
0.7054
0.7389
0.7704
0.7995
0.8264
0.8508
0.8729
0.8925
0.9099
0.9251
0.9382
0.9495
0.9591
0.9671
0.9738
0.9793
0.9838
0.9875
0.9904
0.9927
0.9945
0.9959
0.9969
0.9977
0.9984
0.9988
0.9992
0.9994
0.9996
0.05
0.5199
0.5596
0.5987
0.6368
0.6736
0.7088
0.7422
0.7734
0.8023
0.8289
0.8531
0.8749
0.8944
0.9115
0.9265
0.9394
0.9505
0.9599
0.9678
0.9744
0.9798
0.9842
0.9878
0.9906
0.9929
0.9946
0.9960
0.9970
0.9978
0.9984
0.9989
0.9992
0.9994
0.9996
0.06
0.5239
0.5636
0.6026
0.6406
0.6772
0.7123
0.7454
0.7764
0.8051
0.8315
0.8554
0.8770
0.8962
0.9131
0.9279
0.9406
0.9515
0.9608
0.9686
0.9750
0.9803
0.9846
0.9881
0.9909
0.9931
0.9948
0.9961
0.9971
0.9979
0.9985
0.9989
0.9992
0.9994
0.9996
0.07
0.5279
0.5675
0.6064
0.6443
0.6808
0.7157
0.7486
0.7794
0.8078
0.8340
0.8577
0.8790
0.8980
0.9147
0.9292
0.9418
0.9525
0.9616
0.9693
0.9756
0.9808
0.9850
0.9884
0.9911
0.9932
0.9949
0.9962
0.9972
0.9979
0.9985
0.9989
0.9992
0.9995
0.9996
0.08
0.5319
0.5714
0.6103
0.6480
0.6844
0.7190
0.7517
0.7823
0.8106
0.8365
0.8599
0.8810
0.8997
0.9162
0.9306
0.9429
0.9535
0.9625
0.9699
0.9761
0.9812
0.9854
0.9887
0.9913
0.9934
0.9951
0.9963
0.9973
0.9980
0.9986
0.9990
0.9993
0.9995
0.9996
0.09
0.5359
0.5753
0.6141
0.6517
0.6879
0.7224
0.7549
0.7852
0.8133
0.8389
0.8621
0.8830
0.9015
0.9177
0.9319
0.9441
0.9545
0.9633
0.9706
0.9767
0.9817
0.9857
0.9890
0.9916
0.9936
0.9952
0.9964
0.9974
0.9981
0.9986
0.9990
0.9993
0.9995
0.9997
Safety Stocks
Inventory on hand
I(t)
EOQ
EOQ
order
order
order
ROP
R
mean demand during
supply lead time:
m=RL
Is
I
safety stock s
0
Time t
L
L
L
Comprehensive Financial Evaluation:
Inventory Costs of Palü Gear
1. Cycle Stock (Economies of Scale)
1.1 Optimal order quantity
1.2 # of orders/year
=
=
1.3 Annual ordering cost per store
= $13,009
1.4 Annual cycle stock holding cost.
= $13,009
2. Safety Stock (Uncertainty hedge)
2.1 Safety stock per store
= 70
2.2 Annual safety stock holding cost
= $3,500
3. Total Costs for 5 stores = 5 (13,009 + 13,009 + 3,500)
= 5 x $29,500 = $147.5K.
How to find service level (given ROP)?
How to find re-order point (given SL)?
•
•
•
1.
2.
L
= Supply lead time,
D =N(R, R) = Demand per unit time is normally distributed
with mean R
and
standard deviation R
DL =N(m, ) = Demand during the lead time is normally distributed
with mean m = RL
and
standard deviations   R L
Given ROP, find SL = Cycle service level = P(stock out)
= P(demand during lead time >ROP)
= 1-F(z*= (ROP- m)/)
[use table]
= 1- NORMDIST(ROP, m, , True)
[or Excel]
Given SL, find ROP = m + Is
= m + z* 
[use table to get z* ]
= NORMINV(1-SL, m, )
[or Excel]
Safety stock Is
= z*  Reorder point ROP = m + Is
Take-Aways II
Demand Uncertainty & Inventory Management
Cycle Cost : EOQ: How much to order?
Balance the fixed costs of ordering with the average
inventory holding cost.
Total Cycle Cost are quiet robust to misestimating the
parameters
Growth brings scale economies, adjust operating policies as
markets conditions change.
Safety Stock Cost: Reorder Point : When to place an order?
Uncertainty is nothing but forecasting error.
Determined by the Service Level P(D(lead time)>ROP)
ROP=RxL+Safety Stock
Improving The Supply Chain
How can we reduce supply chain costs without sacrificing
customer service?
How can we improve customer service without increasing
supply chain costs?
Distribution Centralization
Product Postponement (HP)
Process Postponement (Benetton)
Capacity Analysis (Benetton)
Improving Supply Chain Performance:
1. The Effect of Pooling/Centralization
Decentralized Distribution
Is=100
Centralization Distribution
Is=100
Is=400
Is=
400
Is=100
Is=100
Centralized vs. Decentralized Distribution
Pros
Cons
Palü Gear’s Internet restructuring:
Centralized inventory management
 Weekly
demand per store
= 60 jackets/ week
with standard deviation = 30 / week
H = $ 50 / jacket, year S = $ 2,200 / order
Supply lead time L = 2 weeks
Service level : 95% availability.
 Palü
Gear now is considering restructuring to an
Internet store. So 5 local stores will be closed and a
National DC will be opened to distribute direct to
customers.
Compare the safety stock in the decentralized
and centralized systems
Decentralized
Demand R per week
for each store
ms60
s30
Demand during Supplier
lead time (L=2)
mltd
ltd
Safety Stock for each store
(%95 availability)
Centralized – 5 stores
Demand R per week
for the centralized warehouse
mc
c
Demand During Supplier Lead Time
(L=2)
mltd
ltd
Safety Stock for the centralized warehouse
(%95 availability)
Total Safety Stock
Palü Gear’s Internet restructuring:
comprehensive financial inventory evaluation
1. Cycle Stock (Economies of Scale)
1.1 Optimal order quantity
1.2 # of orders/year
=
=
1.3 Annual ordering cost of e-store
= $29,089
1.4 Annual cycle stock holding cost
= $29,089
2. Safety Stock (Uncertainty hedge)
2.1 Safety stock for e-store
=
2.2 Annual safety stock holding cost
= $7,800.
3. Total Costs for consolidated e-store
= 29,089 + 29,089 + 7,800
= $65,980
Learning Objectives:
centralization/pooling
Different methods to achieve pooling efficiencies:
–
–
–
–
Physical centralization
Information centralization
Specialization
Raw material commonality (postponement/late
customization)
 Cost savings are sqrt(# of locations pooled).
Improving Supply Chain Performance:
2. Postponement & Commonality (HP Laserjet)
Generic Power
Production
Unique Power
Production
Transportation
Europe
Process I: Unique
Power Supply
N. America
Europe
Process II: Universal
Power Supply
Make-to-Stock
N. America
Push-Pull Boundary
Make-to-Order
Benetton’s Production and Distribution
Network
Tailored Inventories:
Postponement
• Simple solution
– Produce all garments as Greige goods
(Production cost is 10% higher)
• Tailored solution
– Base load manufactured from colored thread
(cheaper but long lead time sourcing)
– Safety stock held as Greige goods and
manufactured on demand (10% more expensive
but short lead times)
Process Postponement
Dyeing
2 wks
Knitting
2 wks
Make to Stock
Dyeing
2 wks
Dye to Order
Postponement and Re-assortment:
The Advantage to Forecasting
Actual total sales
4000
4000
4000
3500
3500
3500
3000
3000
3000
2500
2500
2500
2000
2000
2000
1500
1500
1500
1000
1000
1000
500
500
500
0
0
0
0
500
1000 1500 2000 2500 3000 3500 4000
Initial Forecast
0
500
1000 1500 2000 2500 3000 3500 4000
0
Updated Forecast after observing 20% of sales
Each data point represents the forecast and the actual
season sales for a particular item (at the style-color level).
500 1000 1500 2000 2500 3000 3500 4000
after 80%
Benetton:
Two types of Production Capacities
Speculative Capacity
Long Lead Time cheap
Commit before
observing the demand
Gamble!
Reactive Capacity
Short Lead Time - expensive
Commit after
observing the early sales data
How do we decide on the size of the speculative
capacity?
Optimal Service Level and Accurate Response to
Demand Uncertainty when you can order only once:
• Palü Gear’s is planning to offer a special line of winter jackets,
especially designed as gifts for the Christmas season. Each
Christma-jacket costs the company $250 and sells for $450. Any
stock left over after Christmas would be disposed of at a deep
discount of $195. Marketing had forecasted a demand of 2000
Christmas-jackets with a forecast error (standard deviation of 500)
How many jacketsDemand
should Palü Gear’s order?
forecast for Christmas jackets
18%
16%
16%
14%
16%
13%
12%
13%
10%
10%
10%
8%
6%
6%
6%
4%
2%
3%
3%
1%
1%
0%
800
1000
1%
1200
1400
1600
1800
2000
2200
2400
2600
2800
3000
1%
3200
Optimal Service Level and Accurate Response: with Excel
(1) Performance for a given order Q, say Q = 2000
Stock:
2000
Demand
Probability
of Demand
units
sold
800
1000
1200
1400
1600
1800
2000
2200
2400
2600
2800
3000
3200
1%
1%
3%
6%
10%
13%
16%
16%
13%
10%
6%
3%
1%
800
1000
1200
1400
1600
1800
2000
2000
2000
2000
2000
2000
2000
1825.6
Expected:
units
units
overstock understock
1200
1000
800
600
400
200
0
0
0
0
0
0
0
141.6
0
0
0
0
0
0
0
200
400
600
800
1000
1200
240.0
Profit
94000
145000
196000
247000
298000
349000
400000
400000
400000
400000
400000
400000
400000
357331
Optimal Service Level and Accurate Response : with Excel
(2) Performance for all possible Q
Order
size Q
Probability
Demand = Q
1000
1001
1200
1400
1600
1800
2000
2200
2400
2600
2800
2999
3000
2%
2%
3%
6%
10%
13%
16%
16%
13%
10%
6%
5%
12%
21%
34%
50%
66%
79%
88%
95%
3%
98%
Cumulative
Expected
Expected
Expected
P (Demand < Q ) units sold units overstock units understock
= F(Q)
983.6
984.6
1177.4
1364.8
1540.2
1696.2
1825.6
1924.0
1991.2
2032.0
2053.3
2062.6
2062.7
0.0
0.0
2.9
12.2
33.6
74.3
141.6
240.0
369.4
525.4
700.8
887.2
888.2
1082.0
1081.0
888.2
700.8
525.4
369.4
240.0
141.6
74.3
33.6
12.2
3.0
2.9
Expected
Profit
$
$
$
$
$
$
$
$
$
$
$
$
$
196,721
196,914
235,323
272,291
306,184
335,142
357,331
371,593
377,929
377,496
372,126
363,726
363,683
 200
  -55
Towards the newsboy model
Suppose you placed an order of 2000 units but you are not sure
about whether you should have ordered one more unit.
What is the contribution of ordering an additional unit?
Marginal Benefit
Earn a margin (p-c) = B with
propability (1-p)= P(D>2000)
P = ..
Marginal Cost
Incur an overage cost of (c-s) = C with
probability p = P(D<=2000)
P = …..
Expected contribution of an additional unit
E(P) = ………………..
So? ... Order more?
Accurate Response:
The newsboy model
In general: at the optimal Q
E(P) <= 0 – no incentives to order more
(1-p)B =
Sell
pC
Do not sell
Equivalently, choose the smallest Q
such that
p = P(D<Q) = F(Q) >= B /
Example:
• Critical fractile
B / (B+C) =
230
220
(B+C)
210
o
200
Expected 190
Profit ($k) 180
170
160
150
140
130
u
Order/Stock
120
Quantity Q
10001200140016001800200022002400260028003000
• Find Q by rounding up!
Q=
- C = - 55
+ 200 = C
Accurate response:
Find optimal Q from newsboy model

Cost of overstocking by one unit = C
– the out-of-pocket cost per unit stocked but not demanded
– “Say demand is one unit below my stock level. How much did the one unit
overstocking cost me?”
E.g.: purchase price - salvage price.

Cost of understocking by one unit = B
– The opportunity cost per unit demanded in excess of the stock level provided
– “Say demand is one unit above my stock level. How much could I have saved
(or gained) if I had stocked one unit more?” E.g.: retail price - purchase price.
Given an order quantity Q, increase it by one unit if and only if the
expected benefit of being able to sell it exceeds the expected cost of having
that unit left over.
Marginal Analysis: Order more as long as F(Q) < B / (B + C)
= smallest Q such that service level F(Q) > critical fractile B / (B + C)
Where else do you find newsboys?
 Benefits:
Flexible Spending Account decision
ATM
 Perishable Products (Newspaper, Medical Supplies,
Fashion Goods)
 Weddings, Conferences…

SUPPLY CHAIN MANGEMENT
Implications:
Economies of Scale
Goal of a Supply Chain
Match Demand with Supply
It is hard … Why?
Manage the trade-off between
Ordering cost and Inventory
Holding Cost
Q*=
2 SR
H
•Growth brings economies of
scale, hence should reflect that
into ordering decisions.
•To reduce the order size n times,
One has to cut the fixed cost per
order by n2 times (the square root
formula!)
Hard to Anticipate Demand
Forecasts are wrong… why?
There is lead time… why there
is lead time?
Lead time (flow time) = Activity
time+ Waiting Time
Because there is waiting time..
Why there is waiting time?
There is inventory in the SC
(Little’s Law)
• As long as you are in the flat
region you are doing fine.
ROP &
Rules of Forecasting
Uncertainty~ Probability Distribution (Mean, SD)
Demand Uncertainty During Lead Time (L)
Shorter Horizon less Uncertainty
Aggregate Forecasts, less uncertain
Why do we hold inventory?
Uncertainty
Forecast Error
Safety Stock Is= zR
L
How do we deal with uncertainty?
Implications:
Is z (service level
appropriate)
Balance overstocking and understocking
Newsboy Problem …
Critical Fractile = 1- P(stockout)
Reduce Lead time
Reduce R
How do we deal with it?
Customer Demand Uncertainty
Normal Variations…
Where does R come from?
Bullwhip Effect
Causes
•Demand Signaling
•Rationing
•Batching
•Promotions
Aggregation
•Physical
•Information
•Specialization
•Component Commonality
•Postponement
How do we deal with it?
•Make the SC more visible
•Align Incentives