Transcript Newsvendor problem and demand uncertainty
Inventory Models
Uncertain Demand: The Newsvendor Model
Background: expected value
A fruit seller example Profit Probability Undamaged mango $ 4 80% Damaged mango $ 1 20% 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: a i probability: p i
Expected value
=
a 1 p 1 + a 2 p 2 + … + a k p k =
S
i = 1,,k a i p i
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
probability
0.05
4 0.12
5 0.20
6 0.24
7 0.17
8 0.14
9 0.08
How many bouquets should he make each morning to
maximize the expected profit
?
Probabilistic models: Flower seller example..
number of bouquets
3
probability
0.05
4 0.12
5 0.20
6 0.24
7 0.17
8 0.14
9 0.08
CASE 1: Make 3 bouquets probability( demand ≥ 3) = 1 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
Probabilistic models: Flower seller example
Compute expected profit for each case
number of bouquets Expected profit
3
probability
0.05
45 4 0.12
57.5
5 0.20
64 6 0.24
60.5
7 0.17
45 8 0.14
21 9 0.08
-10 Making 5 bouquets will maximize expected profit.
Probabilistic models: definitions
number of bouquets probability
3 0.05
4 0.12
5 0.20
6 0.24
7 0.17
8 0.14
9 0.08
Discrete random variable Probability (sum of all likelihoods = 1) Continuous random variable: Example, height of people in a city -4 -3 140 150 160 170 180 190 200 4 Probability density function (area under curve = integral over entire range = 1)
Probabilistic models: normal distribution function
Standard normal distribution curve: mean = 0, std dev. = 1 P( a≤ x ≤ b) = a b
f(x) dx
-4 -3 -2 -1 0 1 2 3 4 a b
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.
The Newsvendor Model
Assumptions: - Plan for single period inventory level - Demand is unknown - p(y) = probability( demand = y), known - Zero setup (ordering) cost
Example: Mrs. Kandell’s Christmas Tree Shop
Order for Christmas trees must be placed in Sept Cost per tree: $25 Price per tree: $55 before Dec 25 $15 after Dec 25 If she orders too few, the
unit shortage cost
is
c u =
55 – 25
=
$30 If she orders too many, the
unit overage cost
is
c o =
25 – 15
=
$10 Past Data Sales Probability 22 .05
24 .10
26 .15
28 .20
30 .20
32 .15
34 .10
36 .05
How many trees should she order?
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
>
Q
stockout, at a cost of:
c u
(
D – Q
) + =
c u
max{
D –Q
, 0}
D
<
Q
overstock, at a cost of
c o
(
Q–D
) +
= c o
max{
Q – D,
0}
Key elements of the model
1. Uncertain demand 2. One chance to order (long) before demand 3. ( order > demand OR order < demand) COST
Model development
Stockout cost =
c u
max{
D –Q
, 0} Overstock cost
= c o
max{
Q – D,
0} Total cost = G(Q) =
c u
(
D – Q
) + +
c o
(
Q – D
) + Expected cost, E( G(Q) ) = E(
c u
(
D – Q
) + +
c o
(
Q – D
) + ) =
c u
E(
D – Q
) + +
c o
E(
Q – D
) +
x
0 [
c u
(
x
Q
)
c o
(
Q
x
) ]
P
(
x
)
x
[
c u Q
(
x
Q
) ]
P
(
x
)
x Q
0 [
c o
(
Q
x
) ]
P
(
x
)
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
E
(
G
(
Q
))
x
Q
[
c u
(
x
Q
) ]
P
(
x
)
x Q
0 [
c o
(
Q
x
) ]
P
(
x
)
g
(
Q
)
E
(
G
(
Q
))
x
Q
0
c
0 (
Q
x
)
P
(
x
)
dx
x
Q c u
(
x
Q
)
P
(
x
)
dx
Model solution
g
(
Q
)
E
(
G
(
Q
))
x Q
0
c
0 (
Q
x
)
P
(
x
)
dx
x
Q c u
(
x
Q
)
P
(
x
)
dx
Minimize g(Q)
d g
(
Q
) 0
dQ d dQ x
Q
0
c
0 (
Q
x
)
P
(
x
)
dx
x
Q c u
(
x
Q
)
P
(
x
)
dx
0 • g(Q) is a convex function: it has a unique minimum • when g(Q) is at minimum value, F(Q) = c u /(c u + c o )
The Critical Ratio Solution to the Newsvendor problem:
dg
(
Q
) 0
dQ
F
(
Q
*)
c
0
c u
c u β = c u /
(
c o + c u
) is called the
critical ratio
b relative importance of
stockout cost
vs.
markdown cost
Mrs. Kandell’s Problem, solved:
c u =
55 – 25
=
$30
c o =
25 – 15
=
$10 Past Data
D
Probability F (D )
22 0.05
0.05
24 0.1
0.15
26 0.15
0.3
28 0.2
0.5
30 0.2
0.7
32 0.15
0.85
34 0.1
0.95
36 0.05
1
β = c u /
(
c o + c u
) = 30/(30 + 10) = 0.75
optimum ≈ 31 NOTE:
E(D) = 22
x
0.05 + 24
x
0.1 + … + 36
x
0.05 = 29
Newsvendor model: effect of critical ratio
D
Probability F (D )
22 0.05
0.05
24 0.1
0.15
26 0.15
0.3
28 0.2
0.5
30 0.2
0.7
32 0.15
0.85
34 0.1
0.95
36 0.05
1
β = c u /
(
c o + c u
) = 30/(30 + 10) = 0.75 optimum: 31 b overstock cost less significant order more b overstock cost dominates order less
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
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
next: Quality Control