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Newsvendor Models

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Newsvendor Models & the Sport Obermeyer Case John H. Vande Vate Fall, 2011 – PowerPoint PPT presentation

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Title: Newsvendor Models


1
Newsvendor Models the Sport Obermeyer Case
  • John H. Vande Vate
  • Fall, 2011

2
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)

3
Finding the Right Mix
  • Managing uncertainty
  • Low cost, high commitment, low flexibility
    (contract)
  • Higher cost, low commitment, higher flexibility
    (spot)

4
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

5
Production Options
  • Hong Kong
  • More expensive
  • Smaller lot sizes
  • Faster
  • More flexible
  • Mainland (Guangdong, Lo Village)
  • Cheaper
  • Larger lot sizes
  • Slower
  • Less flexible

6
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)

7
The Product
  • Gender
  • Styles
  • Colors
  • Sizes
  • Total Number of SKUs 800

8
Service
  • Deliver matching collections simultaneously
  • Deliver early in the season

9
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

10
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
11
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
12
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?

13
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
14
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?

15
Meaningful Objective
  • Maximize the Expected Profit?
  • Maximize Expected ROIC?

16
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
17
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?
18
Question
  • Expected impact
  • P Probability Demand lt Q
  • Reward (Selling Price Cost)(1-P)
  • Risk (Cost Salvage)P
  • How much to order?

19
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?
20
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?

21
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
22
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?

23
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
24
Does this make sense?
Why not do this?
25
P 0.75
  • Explain the strategy
  • Which products are riskier?
  • Which are safer?

26
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?

27
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

28
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
29
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.

30
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?
31
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
32
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

33
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

34
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

35
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?

36
Q as a function of ROIC
Q
ROIC
37
Lets Try It
Min Order Quantities!
38
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

39
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?

40
And China?
38.73 vs 25.5
Min Order Quantities!
41
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
42
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?

43
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!
44
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
45
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?

46
Adding a Lower Bound
Q
ROIC
47
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.

48
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

49
Answers
If everything is made in one place, where would
you make it?
Hong Kong
China
50
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

51
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?

52
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

53
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

54
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

55
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
56
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.
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