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Inventory Models. Uncertain Demand: The Newsvendor Model. 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: a i. probability: p i.

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Presentation Transcript
slide1

Inventory Models

Uncertain Demand: The Newsvendor Model

slide2

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

slide3

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?

slide4

Probabilistic models: Flower seller example..

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

slide5

Probabilistic models: Flower seller example

Compute expected profit for each case 

Making 5 bouquets will maximize expected profit.

slide6

140 150 160 170 180 190 200

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)

slide7

a

b

Probabilistic models: normal distribution function

Standard normal distribution curve: mean = 0, std dev. = 1

P( a≤ x ≤ b) = abf(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.

slide8

The Newsvendor Model

Assumptions:

- Plan for single period inventory level

- Demand is unknown

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

- Zero setup (ordering) cost

slide9

Cost per tree: $25

$55 before Dec 25

$15 after Dec 25

Price per tree:

Example: Mrs. Kandell’s 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?

slide10

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: cu (D – Q)+ = cumax{D –Q, 0}

D < Q overstock, at a cost of co (Q–D)+= comax{Q – D, 0}

slide11

Key elements of the model

1. Uncertain demand

2. One chance to order (long) before demand

3. ( order > demand OR order < demand)  COST

slide12

Expected cost, E( G(Q) ) = E(cu (D – Q)+ + co (Q – D)+)

= cu E(D – Q)+ + co E(Q – D)+

Model development

Stockout cost = cumax{D –Q, 0}

Overstock cost = comax{Q – D, 0}

Total cost = G(Q) = cu (D – Q)+ + co (Q – D)+

slide13

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

slide14

Minimize g(Q) 

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)
slide15

The Critical Ratio

Solution to the Newsvendor problem:

β = cu /(co + cu)is called thecritical ratio

b relative importance of stockout cost vs. markdown cost

slide16

Mrs. Kandell’s Problem, solved:

co = 25 – 15 = $10

cu = 55 – 25 = $30

Past

Data

β = 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

slide17

b

b

Newsvendor model: effect of critical ratio

β = cu /(co + cu)= 30/(30 + 10) = 0.75  optimum: 31

 overstock cost less significant  order more

 overstock cost dominates  order less

slide18

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

slide19

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