The Newsvendor Model: Lecture 10
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Risks from stockout and markdown
The Newsvendor model
Application to postponement
Review for inventory management
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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?
2
Table 1: The Demand
Distribution
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
3
U2’s 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
4
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 =
Overage cost C =
/shirt and
/shirt
5
Fashion Goods
• MBPF and U2’s 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 …
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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?
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The Ordering Risks
• Suppose MBPF or U2 orders Q and demand is D
– If D > 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!
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The Clever Newsboy
How many
papers should
the newsboy
buy?
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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 = B·Prob.{ Demand > Q}
• Expected marginal cost = C·Prob. { Demand ≤ Q}
10
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
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The Critical Ratio
• Suppose Q can be continuous. Then, there is a Q at
which the expected marginal benefit and cost are equal
Prob.( Demand  Q)C  [1  Prob.( Demand  Q)]B
Prob.(Demand  Q ) 
B
BC
(1)
• We call B/(B+C)= β the critical ratio
What does (1) say?
The optimal order quantity Q* is smallest integer
greater than the Q obtained from (1)
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Critical Ratio Solutions
• For MBPF Inc.
B =,
C=
B
P( Demand  Q) 

BC
From MBPFinventory.xls,
Q should be
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Newsvendor with Continuous Demands
•
The demand in the selling cycle can be
characterized by a continuous random variable
D with mean μ, standard deviation σ, and
distribution function F (x)
•
The optimal order quantity Q* is such that
B
F(Q )  Prob.(Demand  Q ) 
BC
*
*
(2)
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Normally Distributed Demands
• Consider normal demands N(μ, σ 2)
with distribution F (Q)
• We then have
Q 
F (Q)  
 B  
   BC
• By this equation, we see that the
critical ratio is the probability
that the standard normal
demand
Ds ≤(Q – μ)/σ.
Prob.(demand≤Q)
µ
Q
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Solution For Normal Demands
• Set (Q – μ)/σ= zβ.
• Recall that there is a one-to-one correspondence
between zβ and β, and they are completely
tabulated in the normal table
• We then have this simple solution:
Q* = μ + zβσ
(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)
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Hedging Factor and Safety Stock
• Hedging factor zβ is a function of the critical ratio β
β 0.1 0.30
0.50
0.75
0.95
0.99
zβ
• When B < C (cost of lost sale < cost of overstock),
overstock is more damaging and we order (zβσ) less than
the expected demand
• When B>C, lost sales is more damaging and we order zβσ
more
• When B=C, the impact of overstock and lost sales are the
same, the best strategy is order the expected demand
• zβσ 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. Park’s past sales record
Sales 29 30
Prob. .05 .10
31
.15
32
.20
33
.20
34
.15
35
.10
36
.05
• Please give: (1) Critical ratio; (2) Hedging factor;
and (3) Safety stock
• Suppose Mrs. Park’s 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 and B
A
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; σ = 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,
retail price  cost
critical ratio   
retail  salvagevalue
Q*    z 
• We will also calculate the expected profit by
( p  c)Q  ( p  s )(Q   )   ( p  s)e
*
*
 z / 2
/ 2.506
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The Value of Postponement
• Option 1: μ= 2000 and σ= 1500
Critical ratio =
postponement
Q*=
Profit from each color =
Total profit =
• Option 2: μ= 8000 and σ= 3000
Critical ratio =
Q*=
Total profit =
25
Value of Postponement with
Dominant Product
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•
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Dominant color: μ=6,200, σ= 4500
Other three colors: μ= 600, σ= 450
Critical ratio =
postponement
Option 1:
Q*1=
profit =
Q*2=
profit =
Total expected profit =
26
Worst off with Postponement
• Option 2:
μ= 8000, σ= (45002+3x4502)1/2 =
4567
Critical ratio =
postponement
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
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Review: Inventory Management
How Much to Order
• Tradeoff between ordering and holding costs
Q*  EOQ  2 RS / H ,
TC (Q* )  2 RSH
TC (Q)  SR / Q  HQ / 2
Robustness and Square-root rule
• Tradeoff between setup time (capacity) and
inventory cost
PQ
RTs P
Qmin 
Pe 
PR
Q  Ts P
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When to Order
• Reorder point ROP =  + IS = RL + zβσ
• Assuming demand is normally distributed:
– For given target SL
ROP =  + zβσ= NORMINV(SL, ,σ)
=  +NORMSINV(SL)·σ
– For given ROP
SL = Pr(DL  ROP) = NORMDIST(ROP, ,.σ, True)
• Safety stock pooling (of n identical locations)
I sa  z  n
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Managing System Inventory
•
•
•
Six basic reasons (functions) to hold inventory
Total average inventory for one item
= Q/2 + zβσ
Not own pipeline
= Q/2 + zβσ+RL
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 B>C, we add a positive safety stock because
stockout is more damaging;
- when B<C, we add a negative safety stock
B
Prob.(Demand  Q) 

BC
• Safety stock is used to hedge the risks
Q* = μ + zβσ
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