Newsvendor problem and demand uncertainty

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Inventory Models
Uncertain Demand The Newsvendor Model
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Background expected value
A fruit seller example
What is the expected profit for a stock of 100
mangoes ?
0.8 x 100 (4) 0.2 x 100 x (1) 320 20
340
random variable ai
probability pi
Expected value a1 p1 a2 p2 ak pk Si
1,,k aipi
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Probabilistic models Flower seller example
Wedding bouquets Selling price 50 (if sold on
same day), 0 (if not sold on that day) Cost
35
How many bouquets should he make each morning to
maximize the expected profit?
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Probabilistic models Flower seller example..
CASE 1 Make 3 bouquets probability( demand 3)
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Exp. Profit 3x50 3x35 45
CASE 2 Make 4 bouquets if demand 3, then
revenue 3x 50 150 if demand 4 or more,
then revenue 4x 50 200
prob 0.05
prob 0.95
Exp. Profit 150x0.05 200x0.95 4x35 57.5
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Probabilistic models Flower seller example
Compute expected profit for each case ?
Making 5 bouquets will maximize expected profit.
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Probabilistic models definitions
Discrete random variable
Probability (sum of all likelihoods 1)
Continuous random variable Example, height of
people in a city
Probability density function (area under curve
integral over entire range 1)
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Probabilistic models normal distribution function
Standard normal distribution curve mean 0, std
dev. 1
P( a x b) ?ab f(x) dx
Property normally distributed random variable
x, mean m, standard deviation
s, Corresponding standard random variable z
(x m)/ s z is normally distributed, with a m
0 and s 1.
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The Newsvendor Model
Assumptions - Plan for single period inventory
level - Demand is unknown - p(y)
probability( demand y), known - Zero setup
(ordering) cost
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Example Mrs. Kandells Christmas Tree Shop
Order for Christmas trees must be placed in Sept
If she orders too few, the unit shortage cost is
cu 55 25 30
If she orders too many, the unit overage cost is
co 25 15 10
Past Data
How many trees should she order?
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Stockout and Markdown Risks
1. Mrs. Kandell has only one chance to
order until the sales begin no information to
revise the forecast after the sales start too
late to order more. 2. She has to decide an
order quantity Q now
D total demand before Christmas F(x) the demand
distribution, D gt Q ? stockout, at a cost
of cu (D Q) cu maxD Q, 0 D lt Q ?
overstock, at a cost of co (QD) co maxQ
D, 0
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Key elements of the model
1. Uncertain demand 2. One chance to order
(long) before demand 3. ( order gt demand OR
order lt demand) ? COST
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Model development
Stockout cost cu maxD Q, 0 Overstock cost
co maxQ D, 0
Total cost G(Q) cu (D Q) co (Q D)
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Model Development generalization
Suppose Demand ? a continuous variable good
approximation when number of possibilities is
high -- difficult to generate probabilities,
but probability distribution can be guessed
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Model solution

• g(Q) is a convex function it has a unique
  minimum
• when g(Q) is at minimum value, F(Q) cu/(cu
  co)

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The Critical Ratio
Solution to the Newsvendor problem
ß cu /(co cu ) is called the critical ratio
b ? relative importance of stockout cost vs.
markdown cost
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Mrs. Kandells Problem, solved
co 25 15 10
cu 55 25 30
Past Data
ß cu /(co cu ) 30/(30 10) 0.75
? optimum 31
NOTE E(D) 22x 0.05 24 x 0.1 36 x
0.05 29
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Newsvendor model effect of critical ratio
ß cu /(co cu ) 30/(30 10) 0.75 ?
optimum 31
? overstock cost less significant ? order more
? overstock cost dominates ? order less
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Summary
When demand is uncertain, we minimize expected
costs newsvendor model single period, with
over- and under-stock costs Critical ratio
determines the optimum order point Critical
ratio affects the direction and magnitude of
order quantity
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Concluding remarks on inventory control
Inventory costs lead to success/failure of a
company
Example Dell Inc. Dell's direct model enables
us to keep low component inventories that enable
us to give customers immediate savings
when component prices are reduced, ... Because of
our inventory management, Dell is able to offer
some of the newest technologies at low prices
while our competitors struggle to sell off older
products.
Drive to reduce inventory costs was main
motivation for Supply Chain Management
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