Betting on Uncertain Demand:
Newsvendor Model
Optional reading: Cachon’s book (reference textbook)
– Ch. 11.
The Newsboy Model: an Example
Mr. Tan, a retiree, sells the local newspaper at
a Bus terminal. At 6:00 am, he meets the
news truck and buys # of the paper at $4.0
and then sells at $8.0. At noon he throws
the unsold and goes home for a nap.
If average daily demand is 50 and he buys
just 50 copies daily, then is the average
daily profit =50*4 =$200?
NO!
Betting on Uncertain Demand
• You must take a firm bet (how much stock
to order) before some random event
occurs (demand) and then you learn that
you either bet too much or too little
• More examples: Products for the
Christmas season; Nokia’s new set,
winter coats, New-Year Flowers, …
Bossini -- Winter Clothes
• Season: Dec. – Jan./Feb.
• Purchase of key materials (fabrics,
dyeing/printing, …) takes long times (upto
90 days)
• Into the selling season, it is too late!
Seattle
Denver
Case: Sport
Obermeyer
Hong
Kong
The SO Supply Chain
Shell Fabric
Lining Fabric
Subcontractors
Insulation mat.
Cut/Sew
Distr Ctr
Retailers
Snaps
Zippers
Others
Textile Suppliers
Obersport
Obermeyer
Retailers
O’Neill’s Hammer 3/2 wetsuit
11-13
Hammer 3/2 timeline and economics
Generate forecast
of demand and
submit an order
to TEC
Spring selling season
Nov Dec Jan
Feb Mar Apr May Jun
Receive order
from TEC at the
end of the
month
Jul Aug
Economics:
•
Each suit sells for
p = $180
•
TEC charges
c = $110 per suit
•
Discounted suits
sell for v = $90
Left over
units are
discounted
• The “too much/too little problem”:
– Order too much and inventory is left over at the end of the
season
– Order too little and sales are lost.
• Marketing’s forecast for sales is 3200 units.
11-14
Newsvendor model
implementation steps
• Gather economic inputs:
– Selling price, production/procurement cost, salvage
value of inventory
• Generate a demand model:
– Use empirical demand distribution or choose a
standard distribution function to represent demand,
e.g. the normal distribution, the Poisson distribution.
• Choose an objective:
– e.g. maximize expected profit or satisfy a fill rate
constraint.
• Choose a quantity to order.
11-15
The Newsvendor Model:
Develop a Forecast
Just one approach
11-16
Historical forecast performance at O’Neill
.
7000
6000
Actual demand
5000
4000
3000
2000
1000
0
0
1000
2000
3000
4000
5000
6000
7000
Forecast
Forecasts and actual demand for surf wet-suits from the previous season
11-17
Actual demand
Product description
Forecast
JR ZEN FL 3/2
90
140
EPIC 5/3 W/HD
120
83
JR ZEN 3/2
140
143
WMS ZEN-ZIP 4/3
170
163
HEATWAVE 3/2
170
212
JR EPIC 3/2
180
175
WMS ZEN 3/2
180
195
ZEN-ZIP 5/4/3 W/HOOD
270
317
WMS EPIC 5/3 W/HD
320
369
EVO 3/2
380
587
JR EPIC 4/3
380
571
WMS EPIC 2MM FULL
390
311
HEATWAVE 4/3
430
274
ZEN 4/3
430
239
EVO 4/3
440
623
ZEN FL 3/2
450
365
HEAT 4/3
460
450
ZEN-ZIP 2MM FULL
470
116
HEAT 3/2
500
635
WMS EPIC 3/2
610
830
WMS ELITE 3/2
650
364
ZEN-ZIP 3/2
660
788
ZEN 2MM S/S FULL
680
453
EPIC 2MM S/S FULL
740
607
EPIC 4/3
1020
732
WMS EPIC 4/3
1060
1552
JR HAMMER 3/2
1220
721
HAMMER 3/2
1300
1696
HAMMER S/S FULL
1490
1832
EPIC 3/2
2190
3504
ZEN 3/2
3190
1195
ZEN-ZIP 4/3
3810
3289
WMS HAMMER 3/2 FULL
6490
3673
* Error = Forecast - Actual demand
** A/F Ratio = Actual demand divided by Forecast
Error* A/F Ratio**
-50
1.56
37
0.69
-3
1.02
7
0.96
-42
1.25
5
0.97
-15
1.08
-47
1.17
-49
1.15
-207
1.54
-191
1.50
79
0.80
156
0.64
191
0.56
-183
1.42
85
0.81
10
0.98
354
0.25
-135
1.27
-220
1.36
286
0.56
-128
1.19
227
0.67
133
0.82
288
0.72
-492
1.46
499
0.59
-396
1.30
-342
1.23
-1314
1.60
1995
0.37
521
0.86
2817
0.57
Probability
Empirical distribution of forecast accuracy
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
A/F ratio
Empirical distribution function for the historical A/F ratios.
How do we know “actual d’d” if it
exceeded forecast?
11-18
Normal distribution tutorial
•
•
•
All normal distributions are characterized by two parameters, mean = m and
standard deviation = s
All normal distributions are related to the standard normal that has mean = 0
and standard deviation = 1.
For example:
– Let Q be the order quantity, and (m, s) the parameters of the normal
demand forecast.
– Prob{demand is Q or lower} = Prob{the outcome of a standard normal
is z or lower}, where
z
Qm
s
or Q  m  z  s
– (The above are two ways to write the same equation, the first allows you
to calculate z from Q and the second lets you calculate Q from z.)
– Look up Prob{the outcome of a standard normal is z or lower} in the
Standard Normal Distribution Function Table.
11-19
Converting between Normal distributions
Start with
m= 100,
s= 25,
Q = 125
0.0180
Center the
distribution over 0
by subtracting the
mean
0.0160
0.0140
0.0120
0.0100
0.0080
0.0060
0.018
0.0040
0.016
0.0020
0.014
0
25
50
75
100
125
150
175
200
0.012
0.01
0.008
0.006
0.004
0.45
z
Qm
s
125 100

25
1
0.002
0.40
0
-100
0.35
-75
-50
-25
0
25
50
75
100
0.30
0.25
0.20
0.15
0.10
0.05
0.00
-4
-3
-2
-1
0
1
2
3
4
Rescale the x and y
axes by dividing by
the standard
deviation
11-20
Using historical A/F ratios to choose a Normal
distribution for the demand forecast
• Start with an initial forecast generated from hunches, guesses, etc.
– O’Neill’s initial forecast for the Hammer 3/2 = 3200 units.
• Evaluate the A/F ratios of the historical data:
A/F ratio 
Actual demand
Forecast
1. Why not just order/buy 3200 units? It
is the most likely outcome!
2. Forecasts always are biased, so
order less than 3200
3. Gross margin is 40%, should order
distribution to more, if is a hit
• Set the mean of the normal
Expectedactualdemand Expected A/F ratio  Forecast
• Set the standard deviation of the normal distribution to
Standarddeviationof actualdem and
Standarddeviationof A/F ratios Forecast
11-21
Actual demand
Product description
Forecast
JR ZEN FL 3/2
90
140
EPIC 5/3 W/HD
120
83
JR ZEN 3/2
140
143
WMS ZEN-ZIP 4/3
170
163
HEATWAVE 3/2
170
212
JR EPIC 3/2
180
175
WMS ZEN 3/2
180
195
ZEN-ZIP 5/4/3 W/HOOD
270
317
WMS EPIC 5/3 W/HD
320
369
EVO 3/2
380
587
JR EPIC 4/3
380
571
WMS EPIC 2MM FULL
390
311
HEATWAVE 4/3
430
274
ZEN 4/3
430
239
EVO 4/3
440
623
ZEN FL 3/2
450
365
HEAT 4/3
460
450
ZEN-ZIP 2MM FULL
470
116
HEAT 3/2
500
635
WMS EPIC 3/2
610
830
WMS ELITE 3/2
650
364
ZEN-ZIP 3/2
660
788
ZEN 2MM S/S FULL
680
453
EPIC 2MM S/S FULL
740
607
EPIC 4/3
1020
732
WMS EPIC 4/3
1060
1552
JR HAMMER 3/2
1220
721
HAMMER 3/2
1300
1696
HAMMER S/S FULL
1490
1832
EPIC 3/2
2190
3504
ZEN 3/2
3190
1195
ZEN-ZIP 4/3
3810
3289
WMS HAMMER 3/2 FULL
6490
3673
* Error = Forecast - Actual demand
** A/F Ratio = Actual demand divided by Forecast
Error* A/F Ratio**
-50
1.56
37
0.69
-3
1.02
7
0.96
-42
1.25
5
0.97
-15
1.08
-47
1.17
-49
1.15
-207
1.54
-191
1.50
79
0.80
156
0.64
191
0.56
-183
1.42
85
0.81
10
0.98
354
0.25
-135
1.27
-220
1.36
286
0.56
-128
1.19
227
0.67
133
0.82
288
0.72
-492
1.46
499
0.59
-396
1.30
-342
1.23
-1314
1.60
1995
0.37
521
0.86
2817
0.57
Probability
Empirical distribution of forecast accuracy
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
A/F ratio
Empirical distribution function for the historical A/F ratios.
11-22
Table 11.2
Actual demand A/F Ratio* Rank Percentile**
Product description
Forecast
ZEN-ZIP 2MM FULL
470
116
0.25
1
3.0%
ZEN 3/2
3190
1195
0.37
2
6.1%
ZEN 4/3
430
239
0.56
3
9.1%
WMS ELITE 3/2
650
364
0.56
4
12.1%
WMS HAMMER 3/2 FULL
6490
3673
0.57
5
15.2%
JR HAMMER 3/2
1220
721
0.59
6
18.2%
HEATWAVE 4/3
430
274
0.64
7
21.2%
ZEN 2MM S/S FULL
680
453
0.67
8
24.2%
EPIC 5/3 W/HD
120
83
0.69
9
27.3%
EPIC 4/3
1020
732
0.72
10
30.3%
WMS EPIC 2MM FULL
390
311
0.80
11
33.3%
ZEN FL 3/2
450
365
0.81
12
36.4%
EPIC 2MM S/S FULL
740
607
0.82
13
39.4%
ZEN-ZIP 4/3
3810
3289
0.86
14
42.4%
WMS ZEN-ZIP 4/3
170
163
0.96
15
45.5%
JR EPIC 3/2
180
175
0.97
16
48.5%
HEAT 4/3
460
450
0.98
17
51.5%
JR ZEN 3/2
140
143
1.02
18
54.5%
WMS ZEN 3/2
180
195
1.08
19
57.6%
WMS EPIC 5/3 W/HD
320
369
1.15
20
60.6%
ZEN-ZIP 5/4/3 W/HOOD
270
317
1.17
21
63.6%
ZEN-ZIP 3/2
660
788
1.19
22
66.7%
HAMMER S/S FULL
1490
1832
1.23
23
69.7%
HEATWAVE 3/2
170
212
1.25
24
72.7%
HEAT 3/2
500
635
1.27
25
75.8%
HAMMER 3/2
1300
1696
1.30
26
78.8%
WMS EPIC 3/2
610
830
1.36
27
81.8%
EVO 4/3
440
623
1.42
28
84.8%
WMS EPIC 4/3
1060
1552
1.46
29
87.9%
JR EPIC 4/3
380
571
1.50
30
90.9%
EVO 3/2
380
587
1.54
31
93.9%
JR ZEN FL 3/2
90
140
1.56
32
97.0%
EPIC 3/2
2190
3504
1.60
33
100.0%
* A/F Ratio = Actual demand divided by Forecast
** Percentile = Rank divided by total number of suits (33)
If the coming year is a similar to the last year,
i.e., the forecasting errors are similar, then,
• There is a 3% chance that demand will be
800 units or fewer (0.25*3200)
• There is a 90.9% chance demand is 150%
of the forecast or lower (or 1.5*3200 = 4,800)
O’Neill’s Hammer 3/2 normal distribution forecast
Product description
JR ZEN FL 3/2
EPIC 5/3 W/HD
JR ZEN 3/2
WMS ZEN-ZIP 4/3
Forecast Actual demand
90
140
120
83
140
143
170
156
Error
-50
37
-3
14
A/F Ratio
1.5556
0.6917
1.0214
0.9176
…
…
…
… …
ZEN 3/2
ZEN-ZIP 4/3
WMS HAMMER 3/2 FULL
Average
Standard deviation
3190
3810
6490
1195
3289
3673
1995
521
2817
0.3746
0.8633
0.5659
0.9975
0.3690
Expectedactualdemand 0.9975 3200 3192
Standarddeviationof actualdemand 0.369  3200 1181
• O’Neill should choose a normal distribution with mean 3192 and
standard deviation 1181 to represent demand for the Hammer 3/2
during the Spring season.
11-25
Empirical vs normal demand distribution
1.00
0.90
Probability
.
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
0
1000
2000
3000
4000
5000
6000
Quantity
Empirical distribution function (diamonds) and normal distribution function with
mean 3192 and standard deviation 1181 (solid line)
11-26
The Newsvendor Model:
The order quantity that
maximizes expected profit
11-27
“Too much” and “too little” costs
• Co = overage cost
– The cost of ordering one more unit than what you would have
ordered had you known demand.
– In other words, suppose you had left over inventory (i.e., you
over ordered). Co is the increase in profit you would have
enjoyed had you ordered one fewer unit.
– For the Hammer 3/2 Co = Cost – Salvage value = c – v = 110 –
90 = 20
• Cu = underage cost
– The cost of ordering one fewer unit than what you would have
ordered had you known demand.
– In other words, suppose you had lost sales (i.e., you under
ordered). Cu is the increase in profit you would have enjoyed had
you ordered one more unit.
– For the Hammer 3/2 Cu = Price – Cost = p – c = 180 – 110 = 70
11-28
Balancing the risk and benefit of ordering a unit
• Ordering one more unit increases the chance of overage …
– Expected loss on the Qth (+1) unit = Co x F(Q)
– F(Q) = Distribution function of demand = Prob{Demand <= Q)
• … but the benefit/gain of ordering one more unit is the reduction in
the chance of underage:
– Expected gain on the Qth (+1) unit = Cu x (1-F(Q))
.
80

70
Expected gain
Expected gain or loss
60
50
40
30
Expected loss
20
As more units are
ordered, the expected
benefit from ordering one
unit decreases while the
expected loss of ordering
one more unit increases.
10
0
0
800
1600
2400
3200
th
4000
Q unit ordered
4800
5600
6400
As we deal with large
numbers, we omit +1
11-29
Newsvendor expected profit
maximizing order quantity
• To maximize expected profit order Q units so that the
expected loss on the Qth unit equals the expected gain
on the Qth unit:
Co  F (Q)  Cu  1  F Q
F (Q) 
• Rearrange terms in the above equation ->
• The ratio Cu / (Co + Cu) is called the critical ratio.
Cu
Co  Cu
• Hence, to maximize profit, choose Q such that we don’t
have lost sales (i.e., demand is Q or lower) with a
probability that equals the critical ratio
11-30
Finding the Hammer 3/2’s expected profit maximizing order
quantity with the empirical distribution function
•
•
Inputs:
– Empirical distribution function table; p = 180; c = 110; v = 90; Cu = 180110 = 70; Co = 110-90 =20
Evaluate the critical ratio:
Cu
70

 0.7778
Co  Cu 20  70
Lookup 0.7778 in the empirical distribution function table
– If the critical ratio falls between two values in the table, choose the one
that leads to the greater order quantity (choose 0.788 which
corresponds to A/F ratio 1.3)
•
…
…
…
170
500
1300
212
635
1696
1.25
1.27
1.30
24
25
26
72.7%
75.8%
78.8%
…
…
Percentile
…
A/F Ratio Rank
…
Actual demand
…
…
HEATWAVE 3/2
HEAT 3/2
HAMMER 3/2
Forecast
…
…
Product description
…
•
Convert A/F ratio into the order quantity
Q  Forecast * A / F  3200 *1.3  4160.
A round-up
rule! See p235.
11-31
Hammer 3/2’s expected profit maximizing order quantity
using the normal distribution
•
Inputs: p = 180; c = 110; v = 90; Cu = 180-110 = 70; Co = 110-90 =20;
critical ratio = 0.7778; mean = m = 3192; standard deviation = s = 1181
•
Look up critical ratio in the Standard Normal Distribution Function Table:
z
0.5
0.6
0.7
0.8
0.9
•
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.6915
0.7257
0.7580
0.7881
0.8159
0.6950
0.7291
0.7611
0.7910
0.8186
0.6985
0.7324
0.7642
0.7939
0.8212
0.7019
0.7357
0.7673
0.7967
0.8238
0.7054
0.7389
0.7704
0.7995
0.8264
0.7088
0.7422
0.7734
0.8023
0.8289
0.7123
0.7454
0.7764
0.8051
0.8315
0.7157
0.7486
0.7794
0.8078
0.8340
0.7190
0.7517
0.7823
0.8106
0.8365
0.7224
0.7549
0.7852
0.8133
0.8389
– If the critical ratio falls between two values in the table, choose the
greater z-statistic
– Choose z = 0.77
Convert the z-statistic into an order quantity: Q  m  z  s
 3192 0.77 1181 4101
11-32