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Inventory Control with
Stochastic Demand
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Lecture Topics
Week 1
Week 2
Week 3
Week 4
Week 5
Week 6
Week 7
Introduction to Production Planning and
Inventory Control
Inventory Control – Deterministic Demand
Inventory Control – Stochastic Demand
Inventory Control – Stochastic Demand
Inventory Control – Stochastic Demand
Inventory Control – Time Varying Demand
Inventory Control – Multiple Echelons
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Lecture Topics (Continued…)
Week 8
Week 9
Week 12
Week 13
Week 14
Week 10
Week 11
Week 15
Production Planning and Scheduling
Production Planning and Scheduling
Managing Manufacturing Operations
Managing Manufacturing Operations
Managing Manufacturing Operations
Demand Forecasting
Demand Forecasting
Project Presentations
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Demand per unit time is a random variable X with mean
E(X) and standard deviation s
Possibility of overstocking (excess inventory) or
understocking (shortages)
There are overage costs for overstocking and shortage
costs for understocking
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Types of Stochastic Models
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)
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Types of Stochastic Models (continued…)
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)
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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
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The Newsvendor Model
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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)
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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
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Notation
X demand (in units), a random variable.
G (x ) P(X x ), cumulative distribution function of demand
(assumed to be continuous)
d
g (x ) G (x ) probability density function of demand.
dx
co cost per unit left over after demand is realized.
cs cost per unit of shortage.
Q Order (or production quantity); a decision variable
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The Cost Function
Y (Q ) expected overage cost + shortage cost
co E units over cs E units short
Q X if Q X
N o Number of units over
otherwise
0
max(Q X , 0) [Q X ]+
X Q if Q X
N S Number of units short
otherwise
0
max( X Q, 0) [ X Q ]
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The Cost Function (Continued…)
Y (Q ) co E[ N o ] cs E[ N S ]
0
0
co max Q x,0 g ( x )dx cs max x Q,0 g ( x )dx
Q
0
Q
co (Q x ) g ( x )dx cs ( x Q ) g ( x )dx
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Leibnitz’s Rule
a2 ( Q )
a2 ( Q )
d
f ( x, Q )dx
[ f ( x, Q )]dx
a
(
Q
)
a
(
Q
)
1
dQ 1
Q
da2 (Q )
da (Q )
f (a2 (Q ), Q )
f (a1 (Q ), Q ) 1
dQ
dQ
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The Optimal Order Quantity
Q
Y (Q )
co g ( x )dx cs g ( x )dx coG (Q ) cs (1 G (Q )) 0
0
Q
Q
G (Q )
cs
co cs
The optimal solution satisfies
G (Q * ) Pr( X Q * )
cs
co cs
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The Exponential Distribution
The Exponential distribution with parameters l
G( x) 1 el x
l e l x ,
g ( x)
0,
1
E( X )
x0
x0
l
Var ( X )
1
l2
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The Exponential Distribution
(Continued…)
G (Q ) 1 e lQ
cs
G (Q*)
cs c0
1 e lQ* Q *
log(1 )
l
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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
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Example (Continued…)
Solution:
G (Q ) 1 e
*
Q
1000
cs
5
0.714
co c s 2 5
Q * 1,253
Sensitivity:
If co = $10 (i.e., shirts must be discarded) then
G (Q * ) 1 e
Q * 405
Q
1000
cs
5
0.333
co c s 10 5
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The Normal Distribution
The Normal distribution with parameters m and s, N(m, s)
1
( x m ) 2
g ( x)
exp[
],
2
2s
s 2
E( X ) m
x
Var ( X ) s 2
• If X has the normal distribution N(m, s), then (X-m)/s has
the standard normal distribution N(0, 1).
• The cumulative distributive function of the Standard
normal distribution is denoted by F.
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The Normal Distribution (Continued…)
G(Q*)=
Pr(X Q*)=
Pr[(X - m)/s (Q* - m)/s] =
Let Y = (X - m)/s, then Y has the the standard Normal
distribution
Pr[(Y (Q* - m)/s] = F[(Q* - m)/s] =
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The Normal Distribution (Continued…)
F((Q* - m)/s) =
Define z such that F(z)
Q* = m + zs
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The Optimal Cost for Normally
Distributed Demand
If Q Q * , then it can be shown that
Y (Q * ) (cs co )s ( z ),
where ( z )
1
z2
exp[
]
2
2
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The Optimal Cost for Normally
Distributed Demand
Both the optimal order quantity and the optimal cost
increase linearly in the standard deviation of demand.
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Example
Demand has the Normal distribution with mean m = 10,000
and standard deviation s = 1,000
cs = 1
co = 0.5 0.67
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Example
Demand has the Normal distribution with mean m = 10,000
and standard deviation s = 1,000
cs = 1
co = 0.5 0.67
Q* = m + zs
From a standard normal table, we find that z0.67 = 0.44
Q* = m + sz 10,000 0.44(1,000) 10,440
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Service Levels
Probability of no stockout
Pr( X Q )
cs
co cs
Fill rate
E[min(Q, X )] E[ X ] E[max( X Q,0)]
E[ N s ]
1
E[ X ]
E[ X ]
E[ X ]
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Service Levels
Probability of no stockout
Pr( X Q )
cs
co cs
Fill rate
E[min(Q, X )] E[ X ] E[max( X Q,0)]
E[ N s ]
1
E[ X ]
E[ X ]
E[ X ]
Fill rate can be significantly higher than
the probability of no stockout
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Discrete Demand
X is a discrete random variable
Y (Q ) co E[ N o ] cs E[ N S ]
co x 0 max Q x,0 Pr( X x ) cs x 0 max x Q,0Pr( X x )
co x 0 (Q x ) Pr( X x ) cs x Q ( x Q ) Pr( X x )
Q
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Discrete Demand (Continued)
The optimal value of Q is the smallest integer that satisfies
Y (Q 1) Y (Q) 0
This is equivalent to choosing the smallest integer Q that
satisfies
x 1 P( X x )
Q
or equivalently
Pr( X Q )
cs
cs co
cs
cs co
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The Geometric Distribution
The geometric distribution with parameter , 0 1
P ( X x ) x (1 ).
E[ X ]
1
Pr( X x ) x
Pr( X x ) 1 x 1
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The Geometric Distribution
The optimal order quantity Q* is the smallest integer that
satisfies
cs
Pr( X Q )
cs co
*
1
Q* 1
cs
Q*
cs co
co
ln[
]
cs co
*
Q
ln[ ]
co
]
cs co
1
ln[ ]
ln[
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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
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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
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Handling Starting Inventory/backorders
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Handling Starting Inventory/backorders
S0 : Starting inventory position
S: order up to level,
S S0 : order quantity
Y ( S ) co E[( S X ) ] cs E[( X S ) ]
cs
The optimal order-up-to level satisfies Pr( X S )
.
cs c0
*
The optimal policy: order nothing if S0 S * , otherwise order S * - S0 .
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