Betting on Uncertain Demand: Newsvendor Model

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!

Hong Kong

Seattle

Denver

Case: Sport Obermeyer

The SO Supply Chain

Shell Fabric

Others

Cut/Sew

Subcontractors Lining Fabric

Insulation mat.

Snaps

Zippers

Distr Ctr Retailers

Textile Suppliers Obersport Obermeyer Retailers

O’Neill’s Hammer 3/2 wetsuit

11-13

Hammer 3/2 timeline and economics

Nov Dec Jan Feb Mar Apr May Jun Jul Aug

Generate forecast of demand and submit an order

to TEC

Receive order from TEC at the

end of the month

Spring selling season

Left overunits are

discounted

Economics:• Each suit sells for

p = $180• TEC charges

c = $110 per suit• Discounted suits

sell for v = $90

• 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.

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The Newsvendor Model:

Develop a Forecast

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Just one approach

Historical forecast performance at O’Neill

0

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7000

0 1000 2000 3000 4000 5000 6000 7000

Forecast

Act

ual d

eman

d

.

Forecasts and actual demand for surf wet-suits from the previous season

11-17

Empirical distribution of forecast accuracy

Empirical distribution function for the historical A/F ratios.

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50%

60%

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90%

100%

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75

A/F ratio

Prob

abili

ty

Product description Forecast Actual demand Error* A/F Ratio**JR ZEN FL 3/2 90 140 -50 1.56EPIC 5/3 W/HD 120 83 37 0.69JR ZEN 3/2 140 143 -3 1.02WMS ZEN-ZIP 4/3 170 163 7 0.96HEATWAVE 3/2 170 212 -42 1.25JR EPIC 3/2 180 175 5 0.97WMS ZEN 3/2 180 195 -15 1.08ZEN-ZIP 5/4/3 W/HOOD 270 317 -47 1.17WMS EPIC 5/3 W/HD 320 369 -49 1.15EVO 3/2 380 587 -207 1.54JR EPIC 4/3 380 571 -191 1.50WMS EPIC 2MM FULL 390 311 79 0.80HEATWAVE 4/3 430 274 156 0.64ZEN 4/3 430 239 191 0.56EVO 4/3 440 623 -183 1.42ZEN FL 3/2 450 365 85 0.81HEAT 4/3 460 450 10 0.98ZEN-ZIP 2MM FULL 470 116 354 0.25HEAT 3/2 500 635 -135 1.27WMS EPIC 3/2 610 830 -220 1.36WMS ELITE 3/2 650 364 286 0.56ZEN-ZIP 3/2 660 788 -128 1.19ZEN 2MM S/S FULL 680 453 227 0.67EPIC 2MM S/S FULL 740 607 133 0.82EPIC 4/3 1020 732 288 0.72WMS EPIC 4/3 1060 1552 -492 1.46JR HAMMER 3/2 1220 721 499 0.59HAMMER 3/2 1300 1696 -396 1.30HAMMER S/S FULL 1490 1832 -342 1.23EPIC 3/2 2190 3504 -1314 1.60ZEN 3/2 3190 1195 1995 0.37ZEN-ZIP 4/3 3810 3289 521 0.86WMS HAMMER 3/2 FULL 6490 3673 2817 0.57* Error = Forecast - Actual demand** A/F Ratio = Actual demand divided by Forecast

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How do we know “actual d’d” if it exceeded forecast?

Normal distribution tutorial

• All normal distributions are characterized by two parameters, mean = and standard deviation =

• 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 (, ) 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

– (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.

orQ

z Q z

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-

0.0020

0.0040

0.0060

0.0080

0.0100

0.0120

0.0140

0.0160

0.0180

0 25 50 75 100 125 150 175 200

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

-100 -75 -50 -25 0 25 50 75 100

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

-4 -3 -2 -1 0 1 2 3 4

Converting between Normal distributions

Start with = 100,= 25, Q = 125

Center the distribution over 0 by subtracting the mean

Rescale the x and y axes by dividing by the standard deviation

125

100125

Q

z

11-20

• 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:

• Set the mean of the normal distribution to

• Set the standard deviation of the normal distribution to

Using historical A/F ratios to choose a Normal distribution for the demand forecast

Forecast

demand Actual ratio A/F

Forecast ratio A/F Expected demand actual Expected

Forecast ratios A/F of deviation Standard

demand actual of deviation Standard

11-21

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 more, if is a hit

Empirical distribution of forecast accuracy

Empirical distribution function for the historical A/F ratios.

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75

A/F ratio

Prob

abili

ty

Product description Forecast Actual demand Error* A/F Ratio**JR ZEN FL 3/2 90 140 -50 1.56EPIC 5/3 W/HD 120 83 37 0.69JR ZEN 3/2 140 143 -3 1.02WMS ZEN-ZIP 4/3 170 163 7 0.96HEATWAVE 3/2 170 212 -42 1.25JR EPIC 3/2 180 175 5 0.97WMS ZEN 3/2 180 195 -15 1.08ZEN-ZIP 5/4/3 W/HOOD 270 317 -47 1.17WMS EPIC 5/3 W/HD 320 369 -49 1.15EVO 3/2 380 587 -207 1.54JR EPIC 4/3 380 571 -191 1.50WMS EPIC 2MM FULL 390 311 79 0.80HEATWAVE 4/3 430 274 156 0.64ZEN 4/3 430 239 191 0.56EVO 4/3 440 623 -183 1.42ZEN FL 3/2 450 365 85 0.81HEAT 4/3 460 450 10 0.98ZEN-ZIP 2MM FULL 470 116 354 0.25HEAT 3/2 500 635 -135 1.27WMS EPIC 3/2 610 830 -220 1.36WMS ELITE 3/2 650 364 286 0.56ZEN-ZIP 3/2 660 788 -128 1.19ZEN 2MM S/S FULL 680 453 227 0.67EPIC 2MM S/S FULL 740 607 133 0.82EPIC 4/3 1020 732 288 0.72WMS EPIC 4/3 1060 1552 -492 1.46JR HAMMER 3/2 1220 721 499 0.59HAMMER 3/2 1300 1696 -396 1.30HAMMER S/S FULL 1490 1832 -342 1.23EPIC 3/2 2190 3504 -1314 1.60ZEN 3/2 3190 1195 1995 0.37ZEN-ZIP 4/3 3810 3289 521 0.86WMS HAMMER 3/2 FULL 6490 3673 2817 0.57* Error = Forecast - Actual demand** A/F Ratio = Actual demand divided by Forecast

11-22

Table 11.2Product description Forecast Actual demand A/F Ratio* Rank Percentile**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

3192320099750 . demand actual Expected

118132003690 . demand actual of deviation Standard

• 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.

Product description Forecast Actual demand Error A/F RatioJR ZEN FL 3/2 90 140 -50 1.5556EPIC 5/3 W/HD 120 83 37 0.6917JR ZEN 3/2 140 143 -3 1.0214WMS ZEN-ZIP 4/3 170 156 14 0.9176

… … … … …ZEN 3/2 3190 1195 1995 0.3746ZEN-ZIP 4/3 3810 3289 521 0.8633WMS HAMMER 3/2 FULL 6490 3673 2817 0.5659Average 0.9975Standard deviation 0.3690

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Empirical vs normal demand distribution

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Quantity

Prob

abili

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Empirical distribution function (diamonds) and normal distribution function withmean 3192 and standard deviation 1181 (solid line)

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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

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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))

0

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70

80

0 800 1600 2400 3200 4000 4800 5600 6400

Qth

unit ordered

Exp

ecte

d ga

in o

r lo

ss

.

Expected gain

Expected loss

As more units are ordered, the expected benefit from ordering one unit decreases while the expected loss of ordering one more unit increases.

11-29

As we deal with large numbers, we omit +1

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:

• Rearrange terms in the above equation ->

• The ratio Cu / (Co + Cu) is called the critical ratio.

• 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

QFCQFC uo 1)(

uo

u

CC

CQF

)(

11-30

Product description Forecast Actual demand A/F Ratio Rank Percentile

… … … … … …

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%

… … … … … …

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 = 180-110 = 70; Co = 110-90 =20

• Evaluate the critical ratio:

• 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)

• Convert A/F ratio into the order quantity

7778.07020

70

uo

u

CC

C

* / 3200 *1.3 4160.Q Forecast A F 11-31

A round-up rule! See p235.

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 = = 3192; standard deviation = = 1181

• Look up critical ratio in the Standard Normal Distribution Function Table:

– 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:

4101118177.03192 zQ

z 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.72240.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.75490.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.78520.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.81330.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389

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