The Newsvendor Model: Lecture 10

1
The Newsvendor Model Lecture 10

• Risks from stockout and markdown
• The Newsvendor model
• Application to postponement
• Review for inventory management

2
Risks from Stockout and Markdown

• MBPF designed a fancy garage FG to sell in the
  Christmas season
• Each costs 3000 in materials and sales for
  5500.
• Unsold FG will be salvaged for 2800 each
• All raw materials have to be purchased in advance
• Based on market research, MBPF estimated the
  demand of FG to be between 10 and 23 and the
  probabilities are given in table 1
• What should be the amount of raw materials to
  purchase for producing FG?

3
Demand Probability
10 0.01
11 0.02
12 0.04
13 0.08
14 0.09
15 0.11
16 0.16
17 0.20
18 0.11
19 0.10
20 0.04
21 0.02
22 0.01
23 0.01
Total 1.00
Table 1 The Demand Distribution
4
U2s Spring T-Shirt

• U2 has a new premier T-Shirt for Spring05 in 4
  colors
• Hong Kong retail market has a 3 month season
  slide 23
• The standard production method is to dye the
  fabric first and then make shirts with different
  colors.
• The production cost is low but leadtime is long,
  at 3 months. So U2 needs to place order in
  December
• The production and in-bound logistic cost is
  30/shirt, and U2 will sell the shirt at
  90/shirt
• U2 does not sell its premier shirts at discount
  in Hong Kong market. After the season, U2
  wholesales the shirts to a mainland company at
  25/shirt

5
Marginal Cost and Marginal Benefit

• Suppose MBPF starts with a potential order
  quantity of Q and considers adding an additional
  unit Q
• - If this unit is sold, there is a benefit
  (profit)
• B
• B is called marginal benefit or underage cost
• - If this unit cannot be sold, there is a cost
• C
• C is called marginal cost or overage cost
• For U2, Underage cost B /shirt and
  Overage cost C /shirt

6
Fashion Goods

• MBPF and U2s have the so called fashion goods
  or newsvendor problem
• Short selling season
• Limited ordering opportunity
• Uncertain demands
• Newspapers, magazines, fish, meat, produce,
  bread, milk, high fashion

7
One Ordering Chance

• MBPF and U2 have only one chance to order (long)
  before the selling season
• Too late to order when the selling starts
• No more demand information before the sales
• There is no way to predict demands accurately
• MBPF keeps past sales record which can be useful
• U2 also can forecast, but what are past sales
  data?

8
The Ordering Risks

• Suppose MBPF or U2 orders Q and demand is D
• If D gt Q, there will be stockout
• The cost (risk) B max D Q, 0
• If D Q, there will be overstocks
• The cost (risk) C max Q D, 0
• The (potential) stockout and markdown costs
• In some industries, such as fashion industry,
  the total stockout and markdown cost is higher
  than the total manufacturing cost!

9
The Clever Newsboy
How many papers should the newsboy buy?
10
The Newsvendor Model

• We do not know for sure if it can be sold or not.
  Thus, we have to work with the expected marginal
  benefit and expected marginal cost
• Expected marginal benefit BProb. Demand gt Q
• Expected marginal cost CProb. Demand Q

11
Marginal Analysis

• Detailed numerical calculations in
  MBPFinventory.xls show, as Q increases
• - The expected marginal benefit decreases
• - The expected marginal cost increases and
• Q 19 is the largest value of Q at which the
  marginal benefit is still greater than the
  marginal cost
• Given an order quantity Q, increase it by one
  unit if and only if the expected benefit of being
  able to sell it exceeds the expected cost of
  having that unit left over

12
The Critical Ratio

• Suppose Q can be continuous. Then, there is a Q
  at which the expected marginal benefit and cost
  are equal
• We call B/(BC) ß the critical ratio
• What does (1) say?
• The optimal order quantity Q is smallest
  integer greater than the Q obtained from (1)

(1)
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Critical Ratio Solutions

• For MBPF Inc.
• B , C
• From MBPFinventory.xls,
• Q should be

14
Newsvendor with Continuous Demands

• The demand in the selling cycle can be
  characterized by a continuous random variable D
  with mean µ, standard deviation s, and
  distribution function F (x)
• The optimal order quantity Q is such that

(2)
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Normally Distributed Demands

• Consider normal demands N(µ, s 2) with
  distribution F (Q)
• We then have
• By this equation, we see that the critical ratio
  is the probability that the standard
  normal demand
  
• Ds (Q µ)/s.

Prob.(demandQ)
µ
Q
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Solution For Normal Demands

• Set (Q µ)/s zß.
• Recall that there is a one-to-one correspondence
  between zß and ß, and they are completely
  tabulated in the normal table
• We then have this simple solution
• Q µ zßs (3)

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Solving Discrete Problems by Normal Approximation

• Consider the product FG of MBPF Inc.
• We use the normal distribution to approximate the
  demand distribution.
• From MBPFinventory.xls
• µ 16.26 and ? 2.48
• From the normal table, we have z0.926
• Then Q
• Also from NORMINV(0.926, 16.26, 2.48)

18
Hedging Factor and Safety Stock

• Hedging factor zß is a function of the critical
  ratio ß
• ß 0.1 0.30 0.50 0.75 0.95 0.99
• zß
• When B lt C (cost of lost sale lt cost of
  overstock), overstock is more damaging and we
  order (zßs) less than the expected demand
• When BgtC, lost sales is more damaging and we
  order zßs more
• When BC, the impact of overstock and lost sales
  are the same, the best strategy is order the
  expected demand
• zßs is called the safety stock

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Exercise Christmas Trees

• Mrs. Park owns a convenience store in Toronto
• Each year, she sells Christmas trees from Dec. 3
  to Dec. 24
• She needs to order the trees in September
• In the season, she sells a tree for 75
• After Dec. 24, an unsold tree is salvaged for 15
• Her cost is 30/tree inclusive

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Exercise Christmas Trees

• Mrs. Parks past sales record
• Sales 29 30 31 32 33 34 35 36
• Prob. .05 .10 .15 .20 .20 .15 .10 .05
• Please give (1) Critical ratio (2) Hedging
  factor and (3) Safety stock
• Suppose Mrs. Parks regular profit margin is 70,
  30, or 10, and all else remain the same. Do the
  same
• christmas

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Postponement

• Delay of product differentiation until closer to
  the time of the sale
• All activities prior to product differentiation
  require aggregate forecasts which are more
  accurate than individual product forecasts

Point of delivery
A
B
A
A and B
B
dyeing
fabricating
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Benefits of Postponement

• Individual product forecasts are only needed
  close to the time of sale demand is known with
  better accuracy (lower uncertainty)
• Results in a better match of supply and demand
• Valuable in e-commerce time lag between when an
  order is placed and when customer receives the
  order (this delay is expected by the customer and
  can be used for postponement)
• Question Is postponement always good? What is
  the main factor(s) that determines the benefits
  of postponement?

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Computing Value of Postponement for U2

• For each color (4 colors) slide 3
• Mean demand µ 2,000 s 1500
• For each garment
• Sale price p 90, Salvage value s 25
• Production cost using Option 1 (long leadtime) c
  30
• Production cost using Option 2 (uncolored
  thread) c 32
• What is the value of postponement?

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Use of The Newsvendor Model

• Recall the newsvendor model,
• We will also calculate the expected profit by

25
The Value of Postponement

• Option 1 µ 2000 and s 1500
• Critical ratio
• Q
• Profit from each color
• Total profit
• Option 2 µ 8000 and s
• Critical ratio
• Q
• Total profit

postponement
3000
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Value of Postponement with Dominant Product

• Dominant color µ6,200, s 4500
• Other three colors µ 600, s 450
• Critical ratio
• Option 1
• Q1 profit
• Q2 profit
• Total expected profit

postponement
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Worst off with Postponement

• Option 2
• µ 8000, s (450023x4502)1/2
• Critical ratio
• Q
• Profit
• Postponement allows a firm to increase profits
  and better match supply and demand if the firm
  produces a large variety of products whose
  demands are not positively correlated and are of
  about the same size

4567
postponement
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Review Inventory Management

• How Much to Order
• Tradeoff between ordering and holding costs
• Robustness and Square-root rule
• Tradeoff between setup time (capacity) and
  inventory cost

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When to Order

• Reorder point ROP ? IS RL zßs
• Assuming demand is normally distributed
• For given target SL
• ROP ? zßs NORMINV(SL, ?,s) ?
  NORMSINV(SL)s
• For given ROP
• SL Pr(DL ? ROP) NORMDIST(ROP, ?,.s, True)
• Safety stock pooling (of n identical locations)

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Managing System Inventory

• Six basic reasons (functions) to hold inventory
• Total average inventory for one item
• Q/2 zßs Not own pipeline
• Q/2 zßsRL Own pipeline
• Managing multiple items
• - ABC analysis 80/20 rule, Pareto Chart

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Newsvendor

• Stockout and markdown are major risks for
  inventory decisions
• The critical ratio balances the stockout cost and
  the markdown cost
• - when BgtC, we add a positive safety stock
  because stockout is more damaging
• - when BltC, we add a negative safety stock
• Safety stock is used to hedge the risks
• Q µ zßs