Inventory Newsvendor

8/12/2019 Inventory Newsvendor

1/19

Uncertain Demand: The Newsvendor Model

Inventory Models


2/19

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


3/19

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?


4/19

Probabilistic models: Flower seller example..


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


5/19

Probabilistic models: Flower seller example


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


6/19

Probabilistic models: definitions


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


7/19

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.


8/19

The Newsvendor Model

Assumptions:

- Plan for single period inventory level

- Demand is unknown

- p(y) = probability( demand = y), known

- Zero setup (ordering) cost


9/19

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


10/19

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


11/19


12/19

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

)(])([)(])([)(])()([


13/19

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


14/19

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


15/19


16/19


17/19

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


18/19

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


19/19

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

Inventory Newsvendor

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