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Newsvendor Models the Sport Obermeyer Case

- John H. Vande Vate
- Fall, 2011

Issues

- Learning Objectives
- Weve 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)

Finding the Right Mix

- Managing uncertainty
- Low cost, high commitment, low flexibility

(contract) - Higher cost, low commitment, higher flexibility

(spot)

Obermeyers Challenge

- Long lead times
- Its November 92 and the company is starting to

make firm commitments for its 93 94 season. - Little or no feedback from market
- First real signal at Vegas trade show in March
- Inaccurate forecasts
- Deep discounts
- Lost sales

Production Options

- Hong Kong
- More expensive
- Smaller lot sizes
- Faster
- More flexible

- Mainland (Guangdong, Lo Village)
- Cheaper
- Larger lot sizes
- Slower
- Less flexible

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)

The Product

- Gender
- Styles
- Colors
- Sizes
- Total Number of SKUs 800

Service

- Deliver matching collections simultaneously
- Deliver early in the season

Production Planning Example

- Rococo Parka
- Wholesale price 112.50
- Average profit 24112.50 27
- Cost 76112.50 85.50
- Average loss (Cost Salvage)
- 8112.50 9
- Salvage (1-24-8)112.50
- (1-32)112.50
- 68112.50
- 76.50

Sample Problem

Forecast is average of the experts forecasts

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

forecasts

Why 2? It has worked

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

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?

ObermeyerData.xls

(1-Margin ) PriceOrder Qty

Margin Price

(1-Margin -Loss ) Price

Profit/Cost

Min(Order Qty, Actual Demand) Price

Max(0, Order Qty-Actual Demand) Price

Revenue Salvage - Cost

Whats the Right Answer?

- There is no right order quantity, we dont know

what demand will be - Whats the right approach to choosing an answer?

Meaningful Objective

- Maximize the Expected Profit?
- Maximize Expected ROIC?

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

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 dont sell it)
- Expected impact?
- Gross ProfitChances we sell last item
- LossChances we dont sell last item
- Expected impact
- P Probability Demand lt Q, the Cycle Service

Level - (Selling Price Cost)(1-P)
- (Cost Salvage)P

Expected reward Why 1-P?

Expected risk Why P?

Question

- Expected impact
- P Probability Demand lt Q
- Reward (Selling Price Cost)(1-P)
- Risk (Cost Salvage)P
- How much to order?

How Much to Order

- Balance the Risks and Rewards
- Reward (Selling Price Cost)(1-P)
- Risk (Cost Salvage)P
- (Selling Price Cost)(1-P) (Cost Salvage)P
- P

If Salvage Value is gt Cost?

How Much to Order

- For Gail
- P
- Selling Price Cost 24Price
- Selling Price Salvage
- Selling Price Cost Cost Salvage
- 24 Price 8Price
- 32 Price
- P 24/32 75
- What does this mean?

For Obermeyer

- Ignoring all other constraints recommended target

Stock Out probability is - 8/(248) 25

Well use 8 of wholesale and 24 of wholesale

across all products

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?

Ignoring Constraints

Everyone has a 25 chance of stockout Everyone

orders Mean 0.6745s

P .75 from .24/(.24.08) Probability of

being less than Mean 0.6745s is 0.75

Does this make sense?

Why not do this?

P 0.75

- Explain the strategy
- Which products are riskier?
- Which are safer?

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?

Invested Capital

- The landed cost (to get product to Obermeyer) is

the investment - Well assume Invested Capital is Cost
- Cost (1-24)Price 76 Price

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

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.

First Phase Objective

Expected Profit Invested Capital

- Maximize ROIC
- Can we exceed a given ROIC?
- Is L(ROIC)
- Max Expected Profit ROICInvested Capital gt 0?

Think of ROIC as an interest payment to

shareholders for the invested capital. Whats the

highest rate of interest we can support?

First Phase Objective

Expected Profit S ciQi

- Maximize ROIC
- Can we achieve return ROIC?
- L(ROIC)
- Max Expected Profit ROIC SciQi gt 0?

The capital ci is the landed cost/unit of

product i

Summary

- Hong Kong
- Cost 76 of Wholesale price
- Profit 24 of Wholesale price
- Salvage Value 68 of Wholesale price
- If we dont 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

Solving for Qi

- For ROIC fixed, how to solve
- L(ROIC) Maximize S Expected Profit(Qi) - ROIC S

ciQi - s.t. Qi ? 0
- Note it is separable (separate decision for each

item) - Exactly the same thinking!
- Last item
- Reward ProfitProbability 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

Hong Kong Solving for Qi

- Last item
- Reward
- (Revenue Cost ROICci)Prob. Demand exceeds Q
- (Revenue Cost ROICci)(1-P)
- Risk
- (Cost ROICci Salvage) Prob. Demand falls

below Q - (Cost ROICci Salvage) P
- As though Cost increased by ROICci , the Tax we

pay to investors

Hong Kong Solving for Qi

- Balance the two
- (Revenue Cost ROICci)(1-P)
- (Cost ROICci Salvage)P
- So P (Profit ROICci)/(Revenue - Salvage)
- Profit/(Revenue - Salvage)

ROICci/(Revenue - Salvage) - In our case
- (Revenue - Salvage) 32 Revenue,
- Profit 24 Revenue
- ci 76 Revenue
- So P 0.75 ROIC76/32 0.75 2.375ROIC
- Recall that P is.
- How does the order quantity Q change with

ROIC?

Q as a function of ROIC

Q

ROIC

Lets Try It

Min Order Quantities!

Summary

- China
- Cost 68.75 of Wholesale price
- Profit 31.25 of Wholesale price
- Salvage Value 68 of Wholesale price
- If we dont 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

In China Solving for Q

- Last item
- Reward (Revenue Cost ROICci)Prob. Demand

exceeds Q - Risk (Cost ROICci Salvage) Prob. Demand

falls below Q - As though Cost increased by ROICci
- Balance the two
- (Revenue Cost ROICci)(1-P) (Cost

ROICci Salvage)P - So P (Profit ROICci)/(Revenue - Salvage)
- Profit/(Revenue - Salvage)

ROICci/(Revenue - Salvage) - In our case
- (Revenue - Salvage) 32 Revenue,
- Profit 31.25 Revenue
- ci 68.75 Revenue
- So P 31.25/32 ROIC68.75/32 0.977

2.148ROIC - Recall that P is.
- How does the order quantity Q change with

ROIC?

And China?

38.73 vs 25.5

Min Order Quantities!

And Minimum Order Quantities

- Maximize S Expected Profit(Qi)
- ROIC SciQi
- Mzi ? Qi ? 600zi (M is a big number)
- zi binary (do we order this or not)

If zi 0 we order 0

If zi 1 we order at least 600

Solving for Qs

- Li(ROIC) Maximize Expected Profit(Qi)

ROICciQi - s.t. Mzi ? Qi ? 600zi
- zi binary
- Two answers to consider
- zi 0 then Li(ROIC) 0
- zi 1 then Qi is easy to calculate
- It is just the larger of 600 and the Q that gives

- P (Profit ROICci)/(Revenue - Salvage)
- (call it Q)
- Which is larger Expected Profit(Q) ROICciQ

or 0?

Which is Larger?

- What is the largest value of ROIC for which,
- Expected Profit(Q) ROICciQ gt 0?
- Expected Profit(Q)/ciQ gt ROIC
- Expected Return on Investment if we make Q is at

least this ROIC - What is this bound?

The return at the minimum order quantity!

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

That integral requires some work

Solving for Qs

- Li(ROIC) Maximize Expected Profit(Qi)
- - ROICciQi
- s.t. Mzi ? Qi ? 600zi
- zi binary
- Lets first look at the problem with zi 1
- Li(ROIC) Maximize Expected Profit(Qi)
- - ROICciQi
- s.t. Qi ? 600
- How does Qi change with ROIC?

Adding a Lower Bound

Q

ROIC

Solving for zi

- Li(ROIC) Maximize Expected Profit(Qi)
- - ROICciQi
- s.t. Mzi ? Qi ? 600zi
- zi binary
- If zi is 0, the objective is 0
- If zi is 1, the objective is
- Expected Profit(Qi) ROICciQi
- So, if Expected Profit(Qi) ROICciQi gt 0, zi is

1 - As 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 ROIC is the return on

600 units.

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, dont make

the product - So, compute the ROIC at the minimum order

quantity and use this to determine when to stop

making the product

Answers

If everything is made in one place, where would

you make it?

Hong Kong

China

Summary

- Simple question of how much to make (no minimums,

no issues of before or after the Vegas show) - Maximize expected profit
- Thats 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

Different View

- Maximize S Expected Profit(Qi)
- S.t. S ci Qi Invested Capital Target
- That maximizes the ROIC for the portfolio
- How to do it?

Different View

- Use Lagrange
- Maximize S Expected Profit(Qi)
- - Tax Rate S 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

Different View

- So What?
- Theres 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

Summary

- Impose minimums (no choice of where to make them)
- If the tax rate exceeds the ROIC at the minimum

order quantity, dont make the product.

Otherwise, make at least the minimum order

quantity - Where to make the product?
- China
- Hong Kong

Where to Produce?

1 if We dont make the product in China and l is

lt Return at 600

If a style is not attractive to produce in China,

it might be attractive in HK at the lower MOQ

Idea

- Its attractive to make it in Hong Kong if
- The return on 1,200 in China is lower than the

tax rate (we dont want to make it there) - but the return on 600 in Hong Kong is higher than

the tax rate (so its still attractive to make it

there) - That doesnt 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.