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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
Recommended