1

Inventory Control with Stochastic Demand

2

Week 1 Introduction to Production Planning and Inventory Control

Week 2 Inventory Control – Deterministic Demand Week 3 Inventory Control – Stochastic Demand Week 4 Inventory Control – Stochastic Demand Week 5 Inventory Control – Stochastic Demand Week 6 Inventory Control – Time Varying Demand Week 7 Inventory Control – Multiple Echelons

Lecture Topics

3

Week 8 Production Planning and Scheduling Week 9 Production Planning and Scheduling Week 12 Managing Manufacturing Operations Week 13 Managing Manufacturing Operations Week 14 Managing Manufacturing Operations Week 10 Demand Forecasting Week 11 Demand Forecasting Week 15 Project Presentations

Lecture Topics (Continued…)

4

Demand per unit time is a random variable X with mean E(X) and standard deviation

Possibility of overstocking (excess inventory) or understocking (shortages)

There are overage costs for overstocking and shortage costs for understocking

5

Single period models Fashion goods, perishable goods, goods with

short lifecycles, seasonal goods One time decision (how much to order)

Multiple period models Goods with recurring demand but whose

demand varies from period to period Inventory systems with periodic review Periodic decisions (how much to order in

each period)

Types of Stochastic Models

6

Continuous time models Goods with recurring demand but with

variable inter-arrival times between customer orders

Inventory system with continuous review Continuous decisions (continuously deciding

on how much to order)

Types of Stochastic Models (continued…)

7

Example

If l is the order replenishment lead time, D is demand per unit time, and r is the reorder point (in a continuous review system), then

Probability of stockout = P(demand during lead time r)

If demand during lead time is normally distributed with mean E(D)l, then choosing r = E(D)l leads to

Probability of stockout = 0.5

8

The Newsvendor Model

9

Assumptions of the Basic Model

A single period

Random demand with known distribution

Cost per unit of leftover inventory (overage cost)

Cost per unit of unsatisfied demand (shortage cost)

10

Objective: Minimize the sum of expected shortage and overage costs

Tradeoff: If we order too little, we incur a shortage cost; if we order too much we incur a an overage cost

11

Notation

demand (in units), a random variable.

( ) ( ), cumulative distribution function of demand

(assumed to be continuous)

( ) ( ) probability density function of demand.

cost per unit left o

X

G x P X x

dg x G x

dxc

over after demand is realized.

cost per unit of shortage.

Order (or production quantity); a decision variablesc

Q

12

The Cost Function

( ) expected overage cost + shortage cost

units over units shorto s

Y Q

c E c E

+

if Number of units over

0 otherwise

max( , 0) [ ]

o

Q X Q XN

Q X Q X

if Number of units short

0 otherwise

max( , 0) [ ]

S

X Q Q XN

X Q X Q

13

The Cost Function (Continued…)

0 0

0

( ) [ ] [ ]

max ,0 ( ) max ,0 ( )

( ) ( ) ( ) ( )

o o s S

o s

Q

o s Q

Y Q c E N c E N

c Q x g x dx c x Q g x dx

c Q x g x dx c x Q g x dx

14

Leibnitz’s Rule

2 2

1 1

( ) ( )

( ) ( )

2 12 1

( , ) [ ( , )]

( ) ( )( ( ), ) ( ( ), )

a Q a Q

a Q a Q

df x Q dx f x Q dx

dQ Q

da Q da Qf a Q Q f a Q Q

dQ dQ

15

The Optimal Order Quantity

0

( )( ) ( ) ( ) (1 ( )) 0

Q

o s o sQ

Y Qc g x dx c g x dx c G Q c G Q

Q

( ) s

o s

cG Q

c c

The optimal solution satisfies

* *( ) Pr( ) s

o s

cG Q X Q

c c

16

The Exponential Distribution

The Exponential distribution with parameters

2

( ) 1

, 0( )

0, 0

1( )

1( )

x

x

G x e

e xg x

x

E X

Var X

17

The Exponential Distribution (Continued…)

0

* *

( ) 1

( *)

log(1 )1

Q

s

s

Q

G Q e

cG Q

c c

e Q

18

Example

Scenario: Demand for T-shirts has the exponential distribution with mean 1000 (i.e., G(x) = P(X x) = 1- e-x/1000)

Cost of shirts is $10. Selling price is $15. Unsold shirts can be sold off at $8.

Model Parameters: cs = 15 – 10 = $5

co = 10 – 8 = $2

19

Example (Continued…)

Solution:

Sensitivity: If co = $10 (i.e., shirts must be discarded)

then

253,1

714.052

51)(

*

1000*

Q

cc

ceQG

so

sQ

405

333.0510

51)(

*

1000*

Q

cc

ceQG

so

sQ

20

The Normal Distribution

The Normal distribution with parameters and , N(, )

• If X has the normal distribution N(, ), then (X-)/ has the standard normal distribution N(0, 1).

•The cumulative distributive function of the Standard normal distribution is denoted by .

2

2

2

1 ( )( ) exp[ ],

22( )

( )

xg x x

E X

Var X

21

The Normal Distribution (Continued…)

G(Q*)=

Pr(X Q*)=

Pr[(X - )/ (Q* - )/] =

Let Y = (X - )/then Y has the the standard Normal distribution

Pr[(Y (Q* - )/] = [(Q* - )/] =

22

The Normal Distribution (Continued…)

((Q* - )/=

Definez such that z)

Q* = + z

23

The Optimal Cost for Normally Distributed Demand

*

*

2

If , then it can be shown that

( ) ( ) ( ),

1where ( ) exp[ ]

22

s o

Q Q

Y Q c c z

zz

24

The Optimal Cost for Normally Distributed Demand

Both the optimal order quantity and the optimal cost increase linearly in the standard deviation of demand.

25

Example

Demand has the Normal distribution with mean = 10,000 and standard deviation = 1,000

cs = 1

co = 0.5

26

Example

Demand has the Normal distribution with mean = 10,000 and standard deviation = 1,000

cs = 1

co = 0.5

Q* = + z

From a standard normal table, we find that z0.67 = 0.44

Q* = + z

27

Service Levels

Probability of no stockout

Fill rate

Pr( ) s

o s

cX Q

c c

[min( , )] [ ] [max( ,0)] [ ]1

[ ] [ ] [ ]sE Q X E X E X Q E N

E X E X E X

28

Service Levels

Probability of no stockout

Fill rate

Fill rate can be significantly higher than the probability of no stockout

Pr( ) s

o s

cX Q

c c

[min( , )] [ ] [max( ,0)] [ ]1

[ ] [ ] [ ]sE Q X E X E X Q E N

E X E X E X

29

Discrete Demand

X is a discrete random variable

0 0

0

( ) [ ] [ ]

max ,0 Pr( ) max ,0 Pr( )

( ) Pr( ) ( ) Pr( )

o o s S

o sx x

Q

o sx x Q

Y Q c E N c E N

c Q x X x c x Q X x

c Q x X x c x Q X x

30

Discrete Demand (Continued)

The optimal value of Q is the smallest integer that satisfies

This is equivalent to choosing the smallest integer Q that satisfies

or equivalently

( 1) ( ) 0Y Q Y Q

1( )

Q sx

s o

cP X x

c c

Pr( ) s

s o

cX Q

c c

31

The Geometric Distribution

1

( ) (1 ).

[ ]1

Pr( )

Pr( ) 1

x

x

x

P X x

E X

X x

X x

The geometric distribution with parameter , 0 1

32

The Geometric Distribution

*

*

1 *

*

Pr( )

ln[ ]

1 1ln[ ]

ln[ ]

ln[ ]

s

s o

o

Q s s o

s o

o

s o

cX Q

c c

cc c c

Qc c

cc c

Q

The optimal order quantity Q* is the smallest integer that satisfies

33

Extension to Multiple Periods

The news-vendor model can be used to a solve a multi-period problem, when:

We face periodic demands that are independent and identically distributed (iid) with distribution G(x)

All orders are either backordered (i.e., met eventually) or lost

There is no setup cost associated with producing an order

34

Extension to Multiple Periods (continued…)

In this case co is the cost to hold one unit of inventory in stock for one period

cs is either the cost of backordering one unit for one period or the cost of a lost sale

35

Handling Starting Inventory/backorders

36

Handling Starting Inventory/backorders

0

0

*

0

0

: Starting inventory position

: order up to level,

: order quantity

( ) [( ) ] [( ) ]

The optimal order-up-to level satisfies Pr( ) .

The optimal policy: order nothing if

o s

s

s

S

S

S S

Y S c E S X c E X S

cX S

c c

S S

* *0, otherwise order - .S S