© Esma Gel, Pınar Keskinocak, 2007
Inventory Management Subject
to Uncertain Demand
ISYE 3104 - Fall 2012
Inventory Control Subject to Uncertain Demand


In the presence of uncertain demand, the objective
is to minimize the expected cost or to maximize the
expected profit
Two types of inventory control models

Fixed time period - Periodic review



One period (Newsvendor model)
Multiple periods
Fixed order quantity - Continuous review

(Q,R) models
1
Types of Inventory Control Policies

Fixed order quantity policies



The order quantity is always the same but the time
between the orders will vary depending on demand
and the current inventory levels
Inventory levels are continuously monitored and an
order is placed whenever the inventory level drops
below a prespecified reorder point.
Continuous review policy
Types of Inventory Control Policies

Fixed time period policies




The time between orders is constant, but the
quantity ordered each time varies with demand and
the current level of inventory
Inventory is reviewed and replenished in given time
intervals, such as a week or month (i.e., review
period)
Depending on the current inventory level an order
size is determined to (possibly) increase the
inventory level up-to a prespecified level (i.e., orderup-to level)
Periodic review policy
2
Inventory Control - Demand
Uncertainty
Stochastic
Deterministic
Variability
Constant/Stationary
Variable/Non-Stationary
Economic Order
Quantity (EOQ) –
Tradeoff between
fixed cost and holding
cost
Lot size/Reorder point
(Q,R) or (s,S) models –
Tradeoff between fixed
cost, holding cost, and
shortage cost
Aggregate Planning –
Planning for capacity
levels given a forecast
Materials Requirements
Planning (MRP)
Very difficult problem!
Newsvendor – single
period
READ THE APPENDIX ON
PROBABILITY REVIEW
3
© Esma Gel, Pınar Keskinocak, 2007
Newsvendor Models
ISYE 3104 – Fall 2012
Example - INFORMS

INFORMS (The Institute for Operations Research and
Management Science, www.informs.org) will hold its annual
meeting in Washington D.C. in 2008. Six months before the
meeting begins, INFORMS must decide how many rooms
should be reserved at the conference hotel. At this time,
rooms can be reserved at a cost of $50 per room. It is
estimated that the demand for rooms is normally distributed
with mean 5000 and standard deviation 2000. If the number
of rooms required exceeds the number of rooms reserved,
extra rooms will have to be found at neighboring hotels at a
cost of $80 per room. The inconvenience of staying at
another hotel is estimated at $10. How many rooms should
be reserved to minimize the expected cost?
4
Example – Fashion Bags

The buyer for What-a-Markup Fashion Bags must
decide on the quantity of a high-priced woman’s
handbag to procure in Italy for the following
Christmas Season. The unit cost of the handbag to
the store is $28.50 and the handbag will sell for
$150. A discount firm purchases any handbags not
sold by the end of the season for $20. In addition,
the store accountants estimate that there is cost of
$0.40 for each dollar tied up in inventory at the end
of the season (after all sales have been made).
Newsvendor model - Properties

One-time decision

Current decisions only impact the “next period” but
not future periods


Retail: fashion/seasonal items
One-time events
5
Tradeoff in inventory decisions
Demand
Supply
Supply < Demand
Shortage
Lost sales / Lost profit
Supply > Demand
Excess inventory
Inventory cost
Supply
Demand
Newsvendor model - Properties

One-time decision


Relevant costs



Co: Unit cost of excess inventory (cost of overage)
Cu: Unit cost of shortage (cost of underage)
Demand D is a random variable with



Current decisions only impact the “next period” but not future
periods
 Retail: fashion/seasonal items
 One-time events
Probability density function (pdf), f(x)
cumulative distribution function (cdf), F(x)
Objective: Choose the ordering quantity Q (before you know
the demand) to minimize

Total expected overage and underage costs
6
Newsvendor model – Finding the optimal order
quantity

First, write the cost function:
G (Q, D ) : total overage  underage cost (if Q is ordered and demand is D)
c ( D  Q ) if D  Q
G (Q, D ) :  u
co (Q  D ) if D  Q
G (Q )  expected overage  underage cost if Q is ordered

G (Q )  E[G (Q, D ]   G (Q, x) f ( x)dx
Critical Ratio
0
Q

0
Q
  co (Q  D) f ( x)dx   cu ( D  Q) f ( x)dx  F (Q ) 
cu
cu  co
Derivation of the Critical Ratio - 1
G (Q )  expected overage  underage cost if Q is ordered
Q

0
Q
G (Q )  co  (Q  D ) f ( x)dx  cu  ( D  Q) f ( x)dx  coG1  cu G2
To take the derivative of G(Q), use Leibniz' s Rule
d
dy
f2 ( y)
 h( x, y)dx 
f1 ( y )
In G1 :
f2 ( y)
h( x, y )
dx  h( f 2 ( y ), y ). f 2' ( y ) h( f1 ( y ), y ). f1' ( y )

y
f1 ( y )

yQ
f 2 ( y)  Q
f1 ( y )  0
h( x, Q)  (Q  x) f ( x)
7
Derivation of the Critical Ratio - 2
Q
G1 (Q)   (Q  D) f ( x)dx
In G1 :
yQ
f 2 ( y)  Q
f1 ( y )  0
h( x, Q)  (Q  x) f ( x)
0
To take the derivative of G(Q), use Leibniz' s Rule
d
dy
f2 ( y)
f2 ( y)
h( x, y )
dx  h( f 2 ( y ), y ). f 2' ( y ) h( f1 ( y ), y ). f1' ( y )
y

f1 ( y )
 h( x, y)dx  
f1 ( y )
h( x, y ) [(Q  D ) f ( x)]

 f ( x)
y
Q
h( f 2 ( y ), y )  h(Q, Q )  (Q  Q) f ( x)  0
f 2' ( y )  1
h( f1 ( y ), y )  h(0, Q)  (Q  0) f ( x)  Qf ( x)
f1' ( y )  0

Q

d
d
G1 (Q )   f ( x)dx Similarly,
G2 (Q)    f ( x)dx
dQ
dQ
Q
0
Derivation of the Critical Ratio - 3
Q

d
G (Q )  co  f ( x)dx  cu  f ( x)dx  co F (Q )  cu (1  F (Q))
dQ
0
Q
 (co  cu ) F (Q)  cu  0
 F (Q) 
cu
cu  co

Recall :
 f ( x)dx  1
0

Q

0
f ( x)dx  F (Q)  P( D  Q )
 f ( x)dx  1  F (Q)  P( D  Q)
Q
Also need to check if G(Q) is convex. Second derivative is (co+cu)f(Q)≥0
8
Example - INFORMS



D = number of rooms actually required
Q = number of rooms reserved
What are Co and Cu ?
Example - INFORMS



D = number of rooms actually required
Q = number of rooms reserved
DQ: cost = 50Q
DQ: cost = 50Q+80(D-Q)+10(D-Q) =90D-40Q
Cost of
Cost of
Cost of
reserving reserving inconvenience
extra rooms
in other hotels
9
Example - INFORMS


D = number of rooms actually required
Q = number of rooms reserved
Co

DQ: cost = 50Q
Cu
DQ: cost = 50Q+80(D-Q)+10(D-Q)=90D-40Q
Cost of
Cost of
Cost of
reserving reserving inconvenience
extra rooms in
other hotels

F (Q ) 
40
4
cu

 
cu  co 40  50 9
What is Q?
Example - INFORMS

Recall: D  Normal(5000,2000)


We have lookup tables for Standard Normal distribution
( =0, =1)  Convert to Standard Normal!

F (Q)  P ( D  Q) 
4
 0.444
9
D Q 
P ( D  Q )  P ( D    Q   )  P

  P( Z  z )  ( z )
 
 
Standard normal

Z
z
Find z from the lookup table  Q =  z + 
10
Example - INFORMS

Recall: D  Normal(5000,2000)


We have lookup tables for Standard Normal distribution
( =0, =1)  Convert to Standard Normal!

F (Q)  P ( D  Q) 
4
 0.444
9
D Q 
P ( D  Q )  P ( D    Q   )  P

  P( Z  z )  ( z )
 
 
Safety
Standard normal

Z
zstock
Expected
demand
Find z from the lookup table  Q =  z + 
Standard Normal Distribution - 1
Density function  ( z ) 
e

z2
2
2
Symmetric around the mean
Area under the entire
curve  1  P(Z  )
11
Standard Normal Distribution - 2
Density function  ( z ) 
e

z2
2
2
Symmetric around the mean
Area under the entire
We will use ɸ(z) and F(z)
interchangeably
curve  1  P(Z  )
F(z)=P(Z≤z)
z
Standard Normal Distribution - 2
Density function  ( z ) 
e

z2
2
2
Symmetric around the mean
Area under the entire
We will use ɸ(z) and F(z)
interchangeably
curve  1  P(Z  )
F(z)=P(Z≤z)
F(-z)=P(Z≤-z)=1-F(z)
-z
z
12
Example - INFORMS
P( Z  z )   ( z )  Area under the curve
 0.444  z  -0.14
Q*   z  
 2000(0.14)  5000
Q*  4720
z = - 0.14
Example - INFORMS
Why is Q*<5000,
i.e., less than the
expected demand?

Expected
total cost
Expected
cost of
underage
Expected
cost of
overage
Q*=4720
13
Example - INFORMS

What if co=150?
Expected
total cost
cu
40

F (Q ) 
cu  co 40  150
 0.21  z  0.805
 Q*  z  
 (2000)(0.805)  5000
 3390
Expected
cost of
overage
Expected
cost of
underage
Q*=3390
Example - INFORMS

What if co=150?
F (Q ) 
cu
40

cu  co 40  150
 0.21  z  0.805
 Q*  z  
 (2000)(0.805)  5000
 3390
Expected
total cost
Co  Q* 
Expected
cost of
overage
Cu  Q* 
Expected
cost of
underage
Q*=3390
14
When do we have Q*=Expected demand?
Q*  z  
If Q*   then z  0 !!!
P( Z  z )  ( z ) 
 Area under the curve
 0.5  z  0
 For Q*   we need
F (Q ) 
cu
 0.5, i.e., cu  co !!!
cu  co
z=0
Optimal quantity versus expected demand
cu  co  F (Q) 
cu
 0.5  z  0  Q*  
cu  co
If shortages are more costly, order more than expected demand
cu  co  F (Q) 
cu
 0.5  z  0  Q*  
cu  co
If excess inventory is more costly order less than expected demand
cu  co  F (Q) 
cu
 0.5  z  0  Q*  
cu  co
If shortages and excess inventory cost the same, order expected demand
Note: Assuming the demand distribution is symmetric around its mean
15
The impact of standard deviation

What happens to the optimal quantity as the
standard deviation increases?
The impact of standard deviation

What happens to the optimal quantity as the standard
deviation increases? (Assuming the demand distribution is symmetric around its mean)
cu  co  z  0  Q*  z   increases in 
If shortages are more costly, Q * increases in std. dev.
cu  co  z  0  Q*  z   decreases in 
If excess inventory is more costly , Q * decreases in std. dev.
cu  co  z  0  Q*   does not change in 
If shortages and excess inventory cost the same,
order expected demand regardless of standard deviation
16
Example - INFORMS
cu=40 < co=50 z=-0.14
cu=70 > co=50 z=0.21
Q*
Q*
Standard deviation
Standard deviation
Example – Fashion Bags
Input




Unit cost
c = $28.50
Selling price
p = $150
Salvage value
s = $20
Cost of inventory = $0.40 for each dollar tied up in
inventory at the end of the season
17
Example – Fashion Bags
Input




Unit cost
c = $28.50
Selling price
p = $150
Salvage value
s = $20
Cost of inventory = $0.40 for each dollar tied up in
inventory at the end of the season
Computed input:



Holding cost per bag
cu =
co =
F (Q) 
h=
cu

cu  co
Example – Fashion Bags
Input




Unit cost
c = $28.50
Selling price
p = $150
Salvage value
s = $20
Cost of inventory = $0.40 for each dollar tied up in
inventory at the end of the season
Computed input:



Holding cost per bag
cu = p – c = $121.5
co = c – s + h = $19.9
F (Q) 
h = (0.40)(28.50) = $11.4
cu
121.5

 0.86
cu  co 121.5  19.9
18
Example – Fashion Bags – Normally distributed
demand
Demand ~ Normal(   150,   20)
F (Q)  0.86  z  1.08
Area under the curve=
Critical ratio = 0.86
 Q*  z  
 (20)(1.08)  150
Q *  172
z = 1.08
Example – Fashion Bags – Uniformly distributed
demand

Demand Uniform between 50 and 250  =150

Same expected demand as in Normal distribution
0.86  (Q * 50)
Q*  222
1

200
Uniform density
Area under the curve =
Critical ratio = 0.86
1/200
Q
250
50
Q*= 222
19
Example – Fashion Bags

Even though both the Normal and the Uniform distributions
have the same mean (=150), why did we get different
quantities?


Normal distribution  Q*=172
Uniform distribution  Q*=222
Example – Fashion Bags

Even though both the Normal and the Uniform distributions
have the same mean (=150), why did we get different
quantities?


Normal distribution  Q*=172
Uniform distribution  Q*=222
Because of the variance (equivalently, standard deviation) and
the shape of the distribution !!!


Normal  =20
Uniform  =57.7
Normal
Uniform
20