Download - Inventory Newsvendor
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Uncertain Demand: The Newsvendor Model
Inventory Models
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Background: expected value
What is the expectedprofit for a stock of 100 mangoes ?
0.8 x 100 ($4) + 0.2 x 100 x ($1) = 320 + 20 = $340
Undamaged mango Damaged mango
Profit $ 4 $ 1
Probability 80% 20%
random variable: ai probability: pi
Expected value= a1p1+ a2p2+ + akpk = Si = 1,,kaipi
A fruit seller example
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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
number of bouquets 3 4 5 6 7 8 9
probability 0.05 0.12 0.20 0.24 0.17 0.14 0.08
How many bouquets should he make each morning
to maximize the expected profit?
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Probabilistic models: Flower seller example..
number of bouquets 3 4 5 6 7 8 9
probability 0.05 0.12 0.20 0.24 0.17 0.14 0.08
CASE 1: Make 3 bouquets
probability( demand 3) = 1 Exp. Profit = 3x503x35 = $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 + 200x0.954x35 = $57.5
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Probabilistic models: Flower seller example
number of bouquets 3 4 5 6 7 8 9
probability 0.05 0.12 0.20 0.24 0.17 0.14 0.08
Expected profit 45 57.5 64 60.5 45 21 -10
Making 5 bouquets will maximize expected profit.
Compute expected profit for each case
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Probabilistic models: definitions
number of bouquets 3 4 5 6 7 8 9
probability 0.05 0.12 0.20 0.24 0.17 0.14 0.08
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)
-4 -3 -2 -1 0 1 2 3
140 150 160 170 180 190 200
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Probabilistic models: normal distribution function
Standard normal distribution curve: mean = 0, std dev. = 1
-4 -3 -2 -1 0 1 2 3
a b
P( a x b) = abf(x) dx
Property:
normally distributed random variable x,
mean = m, standard deviation = s,
Corresponding standard random variable:z = (xm)/ 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
How many trees should she order?
Order for Christmas trees must be placed in Sept
If she orders too few, the unit shortage costis cu= 5525= $30
If she orders too many, the unit overage costis co= 2515= $10
Sales 22 24 26 28 30 32 34 36
Probability .05 .10 .15 .20 .20 .15 .10 .05
Past
Data
Cost per tree: $25 Price per tree:$55 before Dec 25
$15 after Dec 25
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Stockout and Markdown Risks
D total demand before Christmas
F(x) the demand distribution,
D> Q stockout, at a cost of: cu(DQ)+= cumax{DQ, 0}
D< Qoverstock, at a cost of co(QD)+ = comax{QD, 0}
1. Mrs. Kandell has only one chanceto orderuntil 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 Qnow
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Model development
Stockout cost = cumax{DQ, 0}
Overstock cost = comax{QD, 0}
Total cost = G(Q) = cu(DQ)+
+ co(QD)+
Expected cost, E( G(Q) ) = E(cu(DQ)++ co(QD)
+)
= cuE(DQ)+
+ coE(QD)+
Q
x
o
Qx
u
x
ou xPxQcxPQxcxPxQcQxc00
)(])([)(])([)(])()([
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Model Development: generalization
Suppose Demanda continuous variable
++ good approximation when number of possibilities is high
-- difficult to generate probabilities, but
++ probability distribution can be guessed
Q
x
o
Qx
u xPxQcxPQxcQGE0
)(])([)(])([))((
Qx
u
Q
x
dxxPQxcdxxPxQcQGEQg )()()()())(()(0
0
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Model solution
Qx
u
Q
x
dxxPQxcdxxPxQcQGEQg )()()()())(()(0
0
0)()()()(0
0
Qx
u
Q
x
dxxPQxcdxxPxQcdQ
d
g(Q) is a convex function: it has a unique minimum
when g(Q) is at minimum value, F(Q) = cu/(cu+ co)
Minimize g(Q) 0)(
dQ
Qgd
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Newsvendor model: effect of critical ratio
= cu/(co+ cu) = 30/(30 + 10) = 0.75 optimum: 31
D 22 24 26 28 30 32 34 36
Probability 0.05 0.1 0.15 0.2 0.2 0.15 0.1 0.05
F(D) 0.05 0.15 0.3 0.5 0.7 0.85 0.95 1
b overstock cost less significantorder more
b overstock cost dominatesorder 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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Example: Dell I nc.
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.
Concluding remarks on inventory control
Inventory costs lead to success/failure of a company
Drive to reduce inventory costs was main motivation for
Supply Chain Management
next: Quality Control