11

Newsvendor Models & the Sport Obermeyer Case

John H. Vande Vate

Spring, 2012

22

Issues

• Learning Objectives:– We’ve discussed how to measure demand

uncertainty based on historical forecast accuracy

– How to accommodate uncertainty in sourcing• Low cost, high commitment, low flexibility

(“contract”)

• Higher cost, low commitment, higher flexibility (“spot”)

33

Finding the Right Mix

• Managing uncertainty– Low cost, high commitment, low flexibility

(“contract”)– Higher cost, low commitment, higher

flexibility (“spot”)

44

More Generally• Contracts with Carriers

– Assured capacity via contracts– Meet volatile demand with spot

• Labor– Full-time employees – Over-time – Temporary workers

• Capacity– Internal “owned” – Outsourced

• Inventory– Safety Lead Time– Expedited Shipments

55

Obermeyer’s Challenge

• Long lead times:– It’s November ’92 and the company is starting

to make firm commitments for it’s ‘93 – 94 season.

• Little or no feedback from market– First real signal at Vegas trade show in March

• Inaccurate forecasts– Deep discounts– Lost sales

66

Production Options

• Hong Kong– More expensive– Smaller lot sizes– Faster– More flexible

• Mainland (Guangdong, Lo Village)

– Cheaper– Larger lot sizes– Slower– Less flexible

77

The Product

• 5 “Genders”– Price– Type of skier– Fashion quotient

• Example (Adult man)– Fred (conservative, basic)– Rex (rich, latest fabrics and technologies)– Beige (hard core mountaineer, no-nonsense)– Klausie (showy, latest fashions)

88

The Product

• Gender– Styles– Colors– Sizes

• Total Number of SKU’s: ~800

99

Service

• Deliver matching collections simultaneously

• Deliver early in the season

1010

Production Planning Example

• Rococo Parka• Wholesale price $112.50• Average profit 24%*112.50 = $27• Cost = 76%*112.50 = $85.50• Average loss (Cost – Salvage)

– 8%*112.50 = $9• Salvage = (1-24%-8%)*112.50 • = (1-32%)*112.50• = 68%*112.50 • = $76.50

1111

Sample ProblemStyle Price Laura Carolyn Greg Wendy Tom Wally Average Std. Dev 2X Std DevGail 110.00$ 900 1,000 900 1,300 800 1,200 1,017 194 388 Isis 99.00$ 800 700 1,000 1,600 950 1,200 1,042 323 646 Entice 80.00$ 1,200 1,600 1,500 1,550 950 1,350 1,358 248 496 Assault 90.00$ 2,500 1,900 2,700 2,450 2,800 2,800 2,525 340 680 Teri 123.00$ 800 900 1,000 1,100 950 1,850 1,100 381 762 Electra 173.00$ 2,500 1,900 1,900 2,800 1,800 2,000 2,150 404 807 Stephanie 133.00$ 600 900 1,000 1,100 950 2,125 1,113 524 1,048 Seduced 73.00$ 4,600 4,300 3,900 4,000 4,300 3,000 4,017 556 1,113 Anita 93.00$ 4,400 3,300 3,500 1,500 4,200 2,875 3,296 1047 2,094 Daphne 148.00$ 1,700 3,500 2,600 2,600 2,300 1,600 2,383 697 1,394 Total 20,000 20,000 20,000 20,000 20,000 20,000 20,000

Cut and Sew Capacity3000 Units/month

7 month period

First Phase Commitment10,000 units

Second Phase Commitment10,000 units

Individual Forecasts

Forecast is average of the “experts”

forecasts

Std dev of demand about forecast is 2x std dev of forecasts

Why 2? It has worked

1212

Our Approach

• Keep records of Forecast and Actual sales

• Construct a distribution of ratios Actual/Forecast

• Assume next ratio will be a sample from this distribution

Item Forecast Actual Sales Abs Error Error Ratio

1 4349 0 100% -

2 1303 3454 165% 2.65

3 3821 7452 95% 1.95

4 4190 6764 61% 1.61

5 1975 713 64% 0.36

6 4638 4991 8% 1.08

7 1647 519 68% 0.32

8 2454 2030 17% 0.83

9 4567 8210 80% 1.80

10 1747 1350 23% 0.77

11 4824 4572 5% 0.95

12 1628 855 47% 0.53

13 942 1265 34% 1.34

14 3076 1681 45% 0.55

15 2173 2485 14% 1.14

16 1167 743 36% 0.64

17 2983 3388 14% 1.14

18 4746 1512 68% 0.32

19 2408 3163 31% 1.31

20 3126 3643 17% 1.17

21 1000 894 11% 0.89

22 3457 3709 7% 1.07

23 4636 6233 34% 1.34

1313

Distribution of Demand

• We have an estimated distribution of demand (however we get it)

• Example Gail– Mean 1,017 units– Standard deviation 388 units

• Question: How many items to order?

1414

Margin % 24%Loss % 8%

Style Price Profit Salvage Mean Std Dev Order Quantity Revenue Salvage Cost Profit Marginal ROIC Actual DemandGail 110.00$ 26.40$ 74.80$ 1,017 194.08 1000 64,935$ 30,644$ 83,600$ 11,979$ 89% 590.3154

Demand Distribution

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 200 400 600 800 1,000 1,200 1,400 1,600 1,800

Demand

Pro

bab

ility

Get Demand

ObermeyerData.xlsMargin %*

Price

(1-Margin %-Loss %)*Price

(1-Margin %)*Price*Order Qty

Min(Order Qty, Actual Demand)* Price

Max(0, Order Qty-Actual Demand)* Salvage Value

Revenue + Salvage - Cost

Incr. Rev./Cost

1515

What’s the Right Answer?

• There is no “right” order quantity, we don’t know what demand will be

• What’s the right approach to choosing an answer?

1616

Meaningful Objective

• Maximize the Expected Profit?

1717

Marginal ROIC• Marginal Return on Investment:

• Questions: – What happens to Marginal Expected Profit per unit as the order

quantity increases?– What happens to the Marginal Invested Capital as the order

quantity increases?– What happens to Marginal Return on Investment as the order

quantity increases?– What order Quantity maximizes Marginal Return on Investment?– Which styles will show the higher Marginal Return on

Investment?

Marginal Expected Profit

Marginal Invested Capital

1818

Basics: Selecting an Order Quantity

• News Vendor Problem• Order Q• Look at last item, what does it do for us?

Increases our (gross) profits (if we sell it) Increases our losses (if we don’t sell it)

• Expected impact? Gross Profit*Chances we sell last item Loss*Chances we don’t sell last item

• Expected impact P = Probability Demand < Q, the Cycle Service Level (Selling Price – Cost)*(1-P) (Cost – Salvage)*P

Expected reward:

Why 1-P?

Expected risk: Why

P?

1919

Question

• Expected impactP = Probability Demand < QReward: (Selling Price – Cost)*(1-P)Risk: (Cost – Salvage)*P

• How much to order?

2020

How Much to Order

• Balance the Risks and RewardsReward: (Selling Price – Cost)*(1-P)Risk: (Cost – Salvage)*P

(Selling Price – Cost)*(1-P) = (Cost – Salvage)*P

P =

Salvage)– Price (Selling

Cost)– Price (Selling

Salvage)– Cost Cost – Price (Selling

Cost)– Price (Selling

If Salvage Value is >

Cost?

2121

How Much to Order• For Gail:

P =

Selling Price – Cost = 24%Price Selling Price – Salvage = Selling Price – Cost + Cost – Salvage= 24% Price + 8%Price = 32% Price

P = 24/32 = 75%

What does this mean?

Salvage)– Price (Selling

Cost)– Price (Selling

Salvage)– Cost Cost – Price (Selling

Cost)– Price (Selling

2222

For Obermeyer

• Ignoring all other constraints recommended target Stock Out probability is:

= 8%/(24%+8%) = 25%

Salvage) - Price (Selling

Salvage) -(Cost

Salvage) - Price (Selling

Cost)– Price (Selling-1

We’ll use 8% of wholesale and 24% of wholesale across all

products

2323

Simplify our discussion

• Every product has– Gross Profit = 24% of wholesale price– Cost – Salvage = 8% of wholesale price

• Use Normal distribution for demand- Mean is the average forecast- Std dev is 2X the std. dev. of the forecasts

- Every product has recommended P = 0.75

- What does this mean?

2424

Ignoring ConstraintsStyle Mean Std Dev Recommended Order QuantityGail 1,017 388 1,278 Isis 1,042 646 1,478 Entice 1,358 496 1,693 Assault 2,525 680 2,984 Teri 1,100 762 1,614 Electra 2,150 807 2,695 Stephanie 1,113 1048 1,819 Seduced 4,017 1113 4,767 Anita 3,296 2094 4,708 Daphne 2,383 1394 3,323

26,359 Note This suggests over buying!

Everyone has a 25% chance of stockoutEveryone orders Mean + 0.6745

P = .75 [from .24/(.24+.08)] Probability of being less than Mean + 0.6745 is 0.75

2525

Does this make sense?

Style Mean Std Dev Recommended Order QuantityGail 1,017 388 1,278 Isis 1,042 646 1,478 Entice 1,358 496 1,693 Assault 2,525 680 2,984 Teri 1,100 762 1,614 Electra 2,150 807 2,695 Stephanie 1,113 1048 1,819 Seduced 4,017 1113 4,767 Anita 3,296 2094 4,708 Daphne 2,383 1394 3,323

26,359 Note This suggests over buying!

Why not do this?

2626

P = 0.75

• Explain the strategy

• Which products are riskier?• Which are safer?

Style Mean Std Dev Recommended Order QuantityGail 1,017 388 1,278 Isis 1,042 646 1,478 Entice 1,358 496 1,693 Assault 2,525 680 2,984 Teri 1,100 762 1,614 Electra 2,150 807 2,695 Stephanie 1,113 1048 1,819 Seduced 4,017 1113 4,767 Anita 3,296 2094 4,708 Daphne 2,383 1394 3,323

26,359 Note This suggests over buying!

2727

Constraints

• Make at least 10,000 units in initial phase

• Minimum Order Quantities

• What issues should we consider in choosing what to make in the initial phase?

• What objective to consider when choosing what to make in the initial phase?

2828

Invested Capital

• The landed cost (to get product to Obermeyer) is the “investment”

• We’ll assume Invested Capital is Cost

• Cost = (1-24%)*Price = 76% Price

2929

Objective for the “first 10K”

• Invest first in those items with the highest marginal return

• Questions: – What happens to Marginal Expected Profit per unit as the order

quantity increases?– What happens to the Marginal Invested Capital per unit as the

order quantity increases?– What happens to Marginal Return on Investment as the order

quantity increases?– Which styles will show the higher Marginal Return on

Investment?

Marginal Expected Profit

Marginal Invested Capital

3030

Alternative Approach

• Maximize Expected Profits over the season by simultaneously deciding early and late order quantities

• See Fisher and Raman Operations Research 1996

• Requires us to estimate before the Vegas show what our forecasts will be after the show.

3131

First Phase

• Allocate the next units to the SKU with the highest marginal ROIC

• Stop when we’ve allocated all 10,000 units

3232

First Phase Objective:

• ci is the invested capital (cost) per unit

• For a given MROIC

• Max Expected Profit – MROIC ciQi

• The objective is separable

• Max Expected Profit(Qi)-MROIC*ciQi

• Set derivative to 0

• Marginal Expected Profit - MROIC*ci = 0

• MROIC = Marginal Expected Profit

Marginal Invested Capital

3333

First Phase Objective:

• We find Qi so that

• Marginal Expected Profit - MROIC*ci = 0

• = MROIC

• What does this mean about each unit we order?

Marginal Expected Profit(Qi)

Marginal Invested Capital

3434

Solving

• Adjust MROIC until Qi = 10,000

• Why =?

• How to accomplish this?

3535

Ordering

• As though we– Sorted the units of the different skus in

decreasing order of marginal ROIC– Took the top 10,000

3636

Solving for Qi

• For MROIC fixed, how to solveMaximize Expected Profit(Qi) - MROIC ciQi

s.t. Qi 0• Remember it is separable (separate decision for each item)• Exactly the same thinking as the News Vendor• Last item:

– Reward: Profit*Probability Demand exceeds Q– Risk: (Cost – Salvage)* Probability Demand falls below Q– MROIC

• MROIC is like a tax or interest on the investment that adds MROIC * ci to the cost. We pay it whether the item sells or not.

– If it sells, get the original profit – MROIC* ci

– If it doesn’t sell, get (Salvage – Cost – MROIC*ci)

3737

Solving for Qi

• Last item: – Reward:

• (Revenue – Cost – MROIC*ci)*Prob. Demand exceeds Q

• (Revenue – Cost – MROIC*ci)*(1-P)

– Risk: • (Cost + MROIC*ci – Salvage) * Prob. Demand falls below Q

• (Cost + MROIC*ci – Salvage) * P

– As though Cost increased by MROIC*ci , the “Tax” or “Interest” we pay to investors

3838

Hong Kong: Solving for Qi

• Balance the two (Revenue – Cost – MROIC*ci)*(1-P) =

(Cost + MROIC*ci – Salvage)*P

• So P = (Profit – MROIC*ci)/(Revenue - Salvage)

• P = Profit/(Revenue - Salvage) – MROIC*ci/(Revenue - Salvage)

• What happens to P as MROIC increases?• What happens to Qi as MROIC increases?

3939

Summary

• Hong Kong– Cost = 76% of Wholesale price– Profit = 24% of Wholesale price– Salvage Value = 68% of Wholesale price

• If we don’t sell an item, we lose our investment of 76% of wholesale price, but recoup 68% in salvage value. So, net we lose 8% of wholesale price

4040

Hong Kong: Solving for Qi

• So P = (Profit – ROIC*ci)/(Revenue - Salvage)

• = Profit/(Revenue - Salvage) – ROIC*ci/(Revenue - Salvage)

• In our case – (Revenue - Salvage) = 32% Revenue,

– Profit = 24% Revenue

– ci = 76% Revenue

So P = 0.75 – MROIC*76%/32%

= 0.75 – 2.375*MROIC

4141

Q as a function of MROIC

ROIC

Q

0

200

400

600

800

1000

1200

1400

0 0.05 0.1 0.15 0.2 0.25 0.3

4242

Style Mean Std DevRecommended Order Quantity

Wholesale Price Lagrange Order Quantity ROIC

Prob. Demand is less

than 600Largest Return

Gail 1,017 388 1,278 110.00$ 605 25.50% 0.14 25.62%Isis 1,042 646 1,478 99.00$ 356 0.25 21.17%Entice 1,358 496 1,693 80.00$ 833 0.06 28.93%Assault 2,525 680 2,984 90.00$ 1803 0.00 31.48%Teri 1,100 762 1,614 123.00$ 292 0.26 20.81%Electra 2,150 807 2,695 173.00$ 1294 0.03 30.42%Stephanie 1,113 1048 1,819 133.00$ 1 0.31 18.43%Seduced 4,017 1113 4,767 73.00$ 2836 0.00 31.53%Anita 3,296 2094 4,708 93.00$ 1075 0.10 27.41%Daphne 2,383 1394 3,323 148.00$ 905 0.10 27.35%

26,359 10,000

Let’s Try It

Min Order Quantities!

4343

Summary

• China– Cost = 68.75% of Wholesale price– Profit = 31.25% of Wholesale price– Salvage Value = 68% of Wholesale price

• If we don’t sell an item, we lose our investment of 68.75% of wholesale price, but recoup 68% in salvage value. So, net we lose 0.75% of wholesale price

4444

In China: Solving for Q

• So P = (Profit – MROIC*ci)/(Revenue - Salvage)

= Profit/(Revenue - Salvage) – MROIC*ci/(Revenue - Salvage)

• In our case – (Revenue - Salvage) = 32% Revenue, – Profit = 31.25% Revenue

– ci = 68.75% Revenue

So P = 31.25/32 – MROIC*68.75%/32%

= 0.977 – 2.148*MROIC

4545

Style Mean Std Dev

Recommended Order

QuantityWholesale

Price Lagrange Order Quantity MROICGail 1,017 388 1,791 110.00$ 605 38.73%Isis 1,042 646 2,331 99.00$ 356Entice 1,358 496 2,347 80.00$ 833Assault 2,525 680 3,883 90.00$ 1803Teri 1,100 762 2,620 123.00$ 292Electra 2,150 807 3,761 173.00$ 1294Stephanie 1,113 1048 3,203 133.00$ 1Seduced 4,017 1113 6,237 73.00$ 2836Anita 3,296 2094 7,474 93.00$ 1075Daphne 2,383 1394 5,165 148.00$ 905

38,812 10,000

And China?

Min Order Quantities!

38.73% vs 25.5%

Why the same?

4646

And Minimum Order Quantities

In Hong Kong:As we drive up the MROIC, what’s

happening to Qi?

When Qi reaches 600 (the lower bound), what do we know about

Marginal Expected Profit Marginal Investment

What should happen to Qi for values of MROIC higher than this?

4747

Style Mean Std Dev

Recommended Order

QuantityWholesale

Price Lagrange Order Quantity MROICMin Order Quantity

Max Order

Quantity Order?

Return at Minimum

Order Quantity

Gail 1,017 388 1,791 110.00$ 0 34.45% 0 - 0 13.7%Isis 1,042 646 2,331 99.00$ 0 0 - 0 17.7%Entice 1,358 496 2,347 80.00$ 0 0 - 0 28.0%Assault 2,525 680 3,883 90.00$ 2037 1200 3,883 1 44.3%Teri 1,100 762 2,620 123.00$ 0 0 - 0 19.8%Electra 2,150 807 3,761 173.00$ 1570 1200 3,761 1 39.9%Stephanie 1,113 1048 3,203 133.00$ 0 0 - 0 20.6%Seduced 4,017 1113 6,237 73.00$ 3218 1200 6,237 1 45.2%Anita 3,296 2094 7,474 93.00$ 1793 1200 7,474 1 38.1%Daphne 2,383 1394 5,165 148.00$ 1383 1200 5,165 1 36.2%

38,812 10,000

Style Mean Std Dev

Recommended Order

QuantityWholesale

Price Lagrange Order Quantity MROICMin Order Quantity

Max Order

Quantity Order?

Marginal Return at the

Minimum Order

QuantityGail 1,017 388 1,278 110.00$ 641 24.56% 600 1,278 1 25.62%Isis 1,042 646 1,478 99.00$ 0 0 - 0 21.17%Entice 1,358 496 1,693 80.00$ 879 600 1,693 1 28.93%Assault 2,525 680 2,984 90.00$ 1867 600 2,984 1 31.48%Teri 1,100 762 1,614 123.00$ 0 0 - 0 20.81%Electra 2,150 807 2,695 173.00$ 1369 600 2,695 1 30.42%Stephanie 1,113 1048 1,819 133.00$ 0 0 - 0 18.43%Seduced 4,017 1113 4,767 73.00$ 2940 600 4,767 1 31.53%Anita 3,296 2094 4,708 93.00$ 1270 600 4,708 1 27.41%Daphne 2,383 1394 3,323 148.00$ 1035 600 3,323 1 27.35%

26,359 10,000

Answers

China

Hong Kong

If everything is made in one place, where would you make

it?

4848

Summary

• Simple question of how much to make (no minimums, no issues of before or after the Vegas show)– Maximize expected profit

• That’s just a newsvendor problem• Trade off risk of lost sales vs risk of salvage

• Decide which 10,000 to make before show (no minimums, no choice of where to make them)– Highest marginal return on invested capital

4949

Summary

• Impose minimums (no choice of where to make them)– If the tax rate exceeds the MROIC at the

minimum order quantity, don’t make the product. Otherwise, make at least the minimum order quantity

• Where to make the product?– China– Hong Kong

5050

Style Mean Std DevRecommended Order Quantity

Wholesale Price

Order Quantity

Using Lambda MROIC

Min Order Quantity

Max Order

Quantity OrderReturn at Min

Order QuantityGail 1,017 388 1,791 110.00$ 0 34.45% 0 - 0 13.7%Isis 1,042 646 2,331 99.00$ 0 0 - 0 17.7%Entice 1,358 496 2,347 80.00$ 0 0 - 0 28.0%Assault 2,525 680 3,883 90.00$ 2037 1200 3,883 1 44.3%Teri 1,100 762 2,620 123.00$ 0 0 - 0 19.8%Electra 2,150 807 3,761 173.00$ 1570 1200 3,761 1 39.9%Stephanie 1,113 1048 3,203 133.00$ 0 0 - 0 20.6%Seduced 4,017 1113 6,237 73.00$ 3218 1200 6,237 1 45.2%Anita 3,296 2094 7,474 93.00$ 1793 1200 7,474 1 38.1%Daphne 2,383 1394 5,165 148.00$ 1383 1200 5,165 1 36.2%

Gail 1,017 388 1,278 110.00$ 0 0 - 0 25.6%Isis 1,042 646 1,478 99.00$ 0 0 - 0 21.2%Entice 1,358 496 1,693 80.00$ 0 0 - 0 28.9%Assault 2,525 680 2,984 90.00$ 0 0 - 0 31.5%Teri 1,100 762 1,614 123.00$ 0 0 - 0 20.8%Electra 2,150 807 2,695 173.00$ 0 0 - 0 30.4%Stephanie 1,113 1048 1,819 133.00$ 0 0 - 0 18.4%Seduced 4,017 1113 4,767 73.00$ 0 0 - 0 31.5%Anita 3,296 2094 4,708 93.00$ 0 0 - 0 27.4%Daphne 2,383 1394 3,323 148.00$ 0 0 - 0 27.4%

10,000

Same Styles Made in Hong Kong

Where to Produce?

If a style is not attractive to produce in China, it might be attractive in HK at the lower MOQ…

1 if We don’t make the product in

China and MROIC is < Marginal Return at 600

5151

Idea

• It’s attractive to make it in Hong Kong if– The marginal return on 1,200 in China is

lower than the tax rate (we don’t want to make it there)

– but the marginal return on 600 in Hong Kong is higher than the tax rate (so it’s still attractive to make it there)

– That doesn’t happen. – But what if we had to make 20,000 before the

show?

5252

With a 20K Target

Style Mean Std DevRecommended Order Quantity

Wholesale Price

Order Quantity

Using Lambda MROIC

Min Order Quantity

Max Order

Quantity OrderReturn at Min

Order QuantityGail 1,017 388 1,791 110.00$ 0 19.75% 0 - 0 13.7%Isis 1,042 646 2,331 99.00$ 0 0 - 0 17.7%Entice 1,358 496 2,347 80.00$ 1423 1200 2,347 1 28.0%Assault 2,525 680 3,883 90.00$ 2614 1200 3,883 1 44.3%Teri 1,100 762 2,620 123.00$ 1200 1200 2,620 1 19.8%Electra 2,150 807 3,761 173.00$ 2256 1200 3,761 1 39.9%Stephanie 1,113 1048 3,203 133.00$ 1250 1200 3,203 1 20.6%Seduced 4,017 1113 6,237 73.00$ 4163 1200 6,237 1 45.2%Anita 3,296 2094 7,474 93.00$ 3571 1200 7,474 1 38.1%Daphne 2,383 1394 5,165 148.00$ 2566 1200 5,165 1 36.2%

Gail 1,017 388 1,278 110.00$ 791 600 1,278 1 25.6%Isis 1,042 646 1,478 99.00$ 667 600 1,478 1 21.2%Entice 1,358 496 1,693 80.00$ 0 0 - 0 28.9%Assault 2,525 680 2,984 90.00$ 0 0 - 0 31.5%Teri 1,100 762 1,614 123.00$ 0 0 - 0 20.8%Electra 2,150 807 2,695 173.00$ 0 0 - 0 30.4%Stephanie 1,113 1048 1,819 133.00$ 0 0 - 0 18.4%Seduced 4,017 1113 4,767 73.00$ 0 0 - 0 31.5%Anita 3,296 2094 4,708 93.00$ 0 0 - 0 27.4%Daphne 2,383 1394 3,323 148.00$ 0 0 - 0 27.4%

20,502

Same Styles Made in Hong Kong