11

Newsvendor Models & the Sport Obermeyer Case

John H. Vande Vate

Fall, 2011

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

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

55

Production Options

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

• Mainland (Guangdong, Lo Village)

– Cheaper– Larger lot sizes– Slower– Less flexible

66

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)

77

The Product

• Gender– Styles– Colors– Sizes

• Total Number of SKU’s: ~800

88

Service

• Deliver matching collections simultaneously

• Deliver early in the season

99

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

1010

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

1111

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

1212

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?

1313

ObermeyerData.xlsMargin % 24%Loss % 8%

Style Price Profit Salvage Mean Std Dev Order Quantity Revenue Salvage Cost Profit ROICGail 110.00$ 26.40$ 74.80$ 1,017 194.08 1000 110,000$ -$ 83,600$ 26,400$ 32%

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

ba

bili

tyMargin %*

Price(1-Margin %-Loss %)*

Price

(1-Margin %)*Price*Order Qty

Min(Order Qty, Actual Demand)* Price

Max(0, Order Qty-Actual Demand)* Price

Revenue + Salvage - Cost

Profit/Cost

1414

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?

1515

Meaningful Objective

• Maximize the Expected Profit?

• Maximize Expected ROIC?

1616

ROIC

• Return on Investment:

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

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

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

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

Expected Profit

Invested Capital

1717

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?

1818

Question

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

• How much to order?

1919

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?

2020

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

2121

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

2222

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?

2323

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

2424

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?

2525

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!

2626

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?

2727

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

2828

Objective for the “first 10K”

• Return on Investment:

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

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

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

increases?– Which styles will show the higher return on investment?

Expected Profit

Invested Capital

2929

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.

3030

First Phase Objective

• Maximize ROIC =

• Can we exceed a given ROIC*?

• Is L(ROIC*) =

Max Expected Profit – ROIC**Invested Capital > 0?

Expected Profit

Invested Capital

Think of ROIC as an “interest” payment to shareholders for the

invested capital. What’s the highest rate of interest we can

support?

3131

First Phase Objective:

• Maximize ROIC=

• Can we achieve return ROIC?

• L(ROIC) =

Max Expected Profit – ROIC ciQi > 0?

Expected Profit

ciQiThe capital: ci is the landed cost/unit of product i

3232

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

3333

Solving for Qi

• For ROIC fixed, how to solveL(ROIC) = Maximize Expected Profit(Qi) - ROIC ciQi

s.t. Qi 0• Note it is separable (separate decision for each item)• Exactly the same thinking!• Last item:

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

• ROIC is like a tax rate on the investment that adds ROIC * ci to the cost. We pay it whether the item sells or

not

3434

Hong Kong: Solving for Qi

• Last item: – Reward:

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

• (Revenue – Cost – ROIC*ci)*(1-P)

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

• (Cost + ROIC*ci – Salvage) * P

– As though Cost increased by ROIC*ci , the Tax we pay to investors

3535

Hong Kong: Solving for Qi

• Balance the two (Revenue – Cost – ROIC*ci)*(1-P) = (Cost + ROIC*ci – Salvage)*P

• 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 – ROIC*76%/32% = 0.75 – 2.375*ROIC Recall that P is…. How does the order quantity Q change with ROIC?

3636

Q as a function of ROIC

ROIC

Q

0

200

400

600

800

1000

1200

1400

0 0.05 0.1 0.15 0.2 0.25 0.3

3737

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!

3838

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

3939

In China: Solving for Q• Last item:

– Reward: (Revenue – Cost – ROIC*ci)*Prob. Demand exceeds Q– Risk: (Cost + ROIC*ci – Salvage) * Prob. Demand falls below Q– As though Cost increased by ROIC*ci

• Balance the two– (Revenue – Cost – ROIC*ci)*(1-P) = (Cost + ROIC*ci – Salvage)*P

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

– (Revenue - Salvage) = 32% Revenue, – Profit = 31.25% Revenue– ci = 68.75% Revenue

So P = 31.25/32 – ROIC*68.75%/32% = 0.977 – 2.148*ROIC Recall that P is…. How does the order quantity Q change with ROIC?

4040

Style Mean Std Dev

Recommended Order

QuantityWholesale

Price Lagrange Order Quantity lGail 1,017 388 1,278 110.00$ 605 38.73%Isis 1,042 646 1,478 99.00$ 356Entice 1,358 496 1,693 80.00$ 833Assault 2,525 680 2,984 90.00$ 1803Teri 1,100 762 1,614 123.00$ 292Electra 2,150 807 2,695 173.00$ 1294Stephanie 1,113 1048 1,819 133.00$ 1Seduced 4,017 1113 4,767 73.00$ 2836Anita 3,296 2094 4,708 93.00$ 1075Daphne 2,383 1394 3,323 148.00$ 905

26,359 10,000

And China?

Min Order Quantities!

38.73% vs 25.5%

4141

And Minimum Order Quantities

Maximize Expected Profit(Qi)

– ROIC* ciQi

M*zi Qi 600*zi (M is a “big” number)

zi binary (do we order this or not) If zi =1 we

order at least 600

If zi =0 we

order 0

4242

Solving for Q’s

Li(ROIC) = Maximize Expected Profit(Qi) – ROIC*ciQi

s.t. M*zi Qi 600*zi

zi binaryTwo answers to consider:

zi = 0 then Li(ROIC) = 0

zi = 1 then Qi is easy to calculateIt is just the larger of 600 and the Q that gives

P = (Profit – ROIC*ci)/(Revenue - Salvage) (call it Q*)

Which is larger Expected Profit(Q*) – ROIC*ciQ* or 0?

4343

Which is Larger?

• What is the largest value of ROICfor which,

Expected Profit(Q*) – ROIC*ciQ* > 0?

• Expected Profit(Q*)/ciQ* > ROIC

• Expected Return on Investmentif we make Q* is at least this ROIC

• What is this bound? The return at the minimum order

quantity!

4444

Return at Min Order Quantity

• Remember computing the gross profits takes some work, we have to calculate the expected sales

Used a version of the ESC formula to calculate it

600

0( ) 600(1 (600))xf x dx F

That integral requires some work

4545

Solving for Q’s

Li(ROIC) = Maximize Expected Profit(Qi)

- ROIC*ciQi

s.t. M*zi Qi 600*zi

zi binary

Let’s first look at the problem with zi = 1

Li(ROIC) = Maximize Expected Profit(Qi)

- ROIC*ciQi

s.t. Qi 600

How does Qi change with ROIC?

4646

Adding a Lower Bound

0

200

400

600

800

1000

1200

1400

0 0.1 0.2 0.3 0.4

ROIC

Q

0

200

400

600

800

1000

1200

1400

0 0.05 0.1 0.15 0.2 0.25 0.3

4747

Solving for zi

Li(ROIC) = Maximize Expected Profit(Qi) - ROIC*ciQi

s.t. M*zi Qi 600*zi zi binary

If zi is 0, the objective is 0If zi is 1, the objective is

Expected Profit(Qi) – ROIC*ciQi

So, if Expected Profit(Qi) – ROIC*ciQi > 0, zi is 1As we increase the ROIC, Q decreases. Once Q reaches its lower bound, Li(ROIC) decreases, When Li(ROIC) reaches 0, zi changes to 0 and remains 0 Li(ROIC) reaches 0 when ROICis the return on 600 units.

4848

Solving for zi

That was a complicated way of saying that as Q increases, the ROIC decreases

The highest ROIC a product can achieve is the ROIC at its minimum order quantity

If the required ROIC goes above this, don’t make the product

So, compute the ROIC at the minimum order quantity and use this to determine when to stop making the product

4949

Style Mean Std Dev

Recommended Order

QuantityWholesale

Price Lagrange Order Quantity l

Min Order

Quantity

Max Order

Quantity Order?

Return at Minimum

Order Quantity

Gail 1,017 388 1,345 110.00$ 0 37.31% 0 - 0 35.2%Isis 1,042 646 1,588 99.00$ 0 0 - 0 32.1%Entice 1,358 496 1,778 80.00$ 1200 1200 1,778 1 40.5%Assault 2,525 680 3,101 90.00$ 1889 1200 3,101 1 45.2%Teri 1,100 762 1,744 123.00$ 0 0 - 0 31.6%Electra 2,150 807 2,833 173.00$ 1395 1200 2,833 1 43.6%Stephanie 1,113 1048 1,999 133.00$ 0 0 - 0 27.5%Seduced 4,017 1113 4,958 73.00$ 2977 1200 4,958 1 45.4%Anita 3,296 2094 5,067 93.00$ 1339 1200 5,067 1 38.7%Daphne 2,383 1394 3,563 148.00$ 1200 1200 3,563 1 39.5%

27,977 10,000

Style Mean Std Dev

Recommended Order

QuantityWholesale

Price Lagrange Order Quantity lMin Order Quantity

Max Order

Quantity Order?

Return at the Minimum

Order Quantity

Gail 1,017 388 1,278 110.00$ 636 24.71% 600 1,278 1 29.6%Isis 1,042 646 1,478 99.00$ 600 600 1,478 1 24.9%Entice 1,358 496 1,693 80.00$ 872 600 1,693 1 30.6%Assault 2,525 680 2,984 90.00$ 1857 600 2,984 1 31.5%Teri 1,100 762 1,614 123.00$ 0 0 - 0 23.4%Electra 2,150 807 2,695 173.00$ 1357 600 2,695 1 31.0%Stephanie 1,113 1048 1,819 133.00$ 0 0 - 0 16.8%Seduced 4,017 1113 4,767 73.00$ 2924 600 4,767 1 31.6%Anita 3,296 2094 4,708 93.00$ 0 0 - 0 24.7%Daphne 2,383 1394 3,323 148.00$ 1015 600 3,323 1 26.9%

26,359 9,262

Answers

China

Hong Kong

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

it?

5050

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)– Want to ensure a high return on invested capital

5151

Different View

• Maximize Expected Profit(Qi)

• S.t. ci Qi = Invested Capital Target

• That maximizes the ROIC for the “portfolio”

• How to do it?

5252

Different View

• Use Lagrange• Maximize Expected Profit(Qi) • - Tax Rate* ci Qi • At a given Tax Rate, the answer maximizes the

ROIC over all portfolios with that amount of Invested Capital.

• Increasing the Tax Rate reduces the Invested Capital

• So, we can carve out the frontier of high ROIC portfolios vs Invested Capital

5353

Different View

• So What?

• There’s no constraint on Invested Capital

• There is a target for total units – 10,000

• Adjust the Tax Rate until we find a high ROIC portfolio with close to 10,000 units

5454

Summary

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

5555

Where to Produce?

Style Mean Std DevRecommended Order Quantity

Wholesale Price

Order Quantity

Using Lambda l

Min Order

Quantity

Max Order

Quantity Order

Return at Min Order

QuantityGail 1,017 388 1,345 110.00$ 0 37.31% 0 - 0 35.2%Isis 1,042 646 1,588 99.00$ 0 0 - 0 32.1%Entice 1,358 496 1,778 80.00$ 1200 1200 1,778 1 40.5%Assault 2,525 680 3,101 90.00$ 1889 1200 3,101 1 45.2%Teri 1,100 762 1,744 123.00$ 0 0 - 0 31.6%Electra 2,150 807 2,833 173.00$ 1395 1200 2,833 1 43.6%Stephanie 1,113 1048 1,999 133.00$ 0 0 - 0 27.5%Seduced 4,017 1113 4,958 73.00$ 2977 1200 4,958 1 45.4%Anita 3,296 2094 5,067 93.00$ 1339 1200 5,067 1 38.7%Daphne 2,383 1394 3,563 148.00$ 1200 1200 3,563 1 39.5%

Gail 1,017 388 1,278 110.00$ 0 0 - 0 29.6%Isis 1,042 646 1,478 99.00$ 0 0 - 0 24.9%Entice 1,358 496 1,693 80.00$ 0 0 - 0 30.6%Assault 2,525 680 2,984 90.00$ 0 0 - 0 31.5%Teri 1,100 762 1,614 123.00$ 0 0 - 0 23.4%Electra 2,150 807 2,695 173.00$ 0 0 - 0 31.0%Stephanie 1,113 1048 1,819 133.00$ 0 0 - 0 16.8%Seduced 4,017 1113 4,767 73.00$ 0 0 - 0 31.6%Anita 3,296 2094 4,708 93.00$ 0 0 - 0 24.7%Daphne 2,383 1394 3,323 148.00$ 0 0 - 0 26.9%

10,000

Same Styles Made in Hong Kong

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 l is < Return at 600

5656

Idea

• It’s attractive to make it in Hong Kong if– The return on 1,200 in China is lower than the tax rate

(we don’t want to make it there)– but the 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. We always get a higher return on

1,200 in China than on 600 in HK– In fact the lowest return on 1200 in China is greater

than the highest return on 600 in HK.

• Conclusion: Only use HK after the Vegas show for small volume products.