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Inventory Models
Inventory Models
Background: expected value
Background: expected value
Probabilistic models: Flower seller example
Probabilistic models: Flower seller example
Probabilistic models: Flower seller example
Probabilistic models: Flower seller example
Probabilistic models: Flower seller example
Probabilistic models: Flower seller example
Probabilistic models: definitions
Probabilistic models: definitions
Probabilistic models: normal distribution function
Probabilistic models: normal distribution function
The Newsvendor Model
The Newsvendor Model
Example: Mrs
Example: Mrs
Stockout and Markdown Risks
Stockout and Markdown Risks
Key elements of the model
Key elements of the model
Model development
Model development
Model Development: generalization
Model Development: generalization
Model solution
Model solution
The Critical Ratio
The Critical Ratio
Mrs
Mrs
Newsvendor model: effect of critical ratio
Newsvendor model: effect of critical ratio
Summary
Summary
Concluding remarks on inventory control
Concluding remarks on inventory control

Презентация на тему: «Uncertain Demand: The Newsvendor Model». Автор: Liu Liming. Файл: «Uncertain Demand: The Newsvendor Model.ppt». Размер zip-архива: 121 КБ.

Uncertain Demand: The Newsvendor Model

содержание презентации «Uncertain Demand: The Newsvendor Model.ppt»
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1 Inventory Models

Inventory Models

Uncertain Demand: The Newsvendor Model

2 Background: expected value

Background: expected value

A fruit seller example

Undamaged mango

Damaged mango

Profit

$ 4

$ 1

Probability

80%

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: ai

probability: pi

Expected value = a1 p1 + a2 p2 + … + ak pk = Si = 1,,k aipi

3 Probabilistic models: Flower seller example

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 Probabilistic models: Flower seller example

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 = 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

5 Probabilistic models: Flower seller example

Probabilistic models: Flower seller example

Compute expected profit for each case ?

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.

6 Probabilistic models: definitions

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)

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

7 Probabilistic models: normal distribution function

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.

8 The Newsvendor Model

The Newsvendor Model

Assumptions: - Plan for single period inventory level - Demand is unknown - p(y) = probability( demand = y), known - Zero setup (ordering) cost

9 Example: Mrs

Example: Mrs

Kandell’s Christmas Tree Shop

Order for Christmas trees must be placed in Sept

How many trees should she order?

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

Sales

22

24

26

28

30

32

34

36

Probability

.05

.10

.15

.20

.20

.15

.10

.05

10 Stockout and Markdown Risks

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)+ = cu max{D –Q, 0} D < Q ? overstock, at a cost of co (Q–D)+ = co max{Q – D, 0}

11 Key elements of the model

Key elements of the model

1. Uncertain demand 2. One chance to order (long) before demand 3. ( order > demand OR order < demand) ? COST

12 Model development

Model development

Stockout cost = cu max{D –Q, 0} Overstock cost = co max{Q – D, 0}

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

13 Model Development: generalization

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

14 Model solution

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)

15 The Critical Ratio

The Critical Ratio

Solution to the Newsvendor problem:

? = cu /(co + cu ) is called the critical ratio

b ? relative importance of stockout cost vs. markdown cost

16 Mrs

Mrs

Kandell’s Problem, solved:

co = 25 – 15 = $10

cu = 55 – 25 = $30

? = cu /(co + cu ) = 30/(30 + 10) = 0.75

Past Data

? optimum ? 31

NOTE: E(D) = 22x 0.05 + 24 x 0.1 + … + 36 x 0.05 = 29

17 Newsvendor model: effect of critical ratio

Newsvendor model: effect of critical ratio

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

? overstock cost less significant ? order more

? overstock cost dominates ? order less

18 Summary

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 Concluding remarks on inventory control

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

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.”

next: Quality Control

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