MS 401
Production and Service Systems Operations
Spring 2009-2010
Inventory Control – III
Stochastic Demand: Newsboy Model
Slide Set #5
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Newsboy’s Problem
•
•
•
•
A newsboy is selling the magazine Atlas
The newsboy buys each copy for $3.00 and sells it for $4.00
At end of the week he can return each unsold copy for $2.5
Question: How many copies to buy at the beginning of the
week?
observed
demand in the
past 52-week
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15
14
8
6
8
13
19 9 12 9 22 4 7 8
11 6 11 9 18 10 1 14
9 5 4 4 17 18 14 15
7 12 15 15 19 9 10 9
11 11 18 15 17 19 14 14
12
11
12
8
16
17
2
Newsboy Example
• Normal approximation to demand:
mean: 11.73,
standard deviation: 4.74
• Perhaps, the newsboy should buy 12 magazines to satisfy
the average demand?
• The cost for each issue unsold (cost of overage):
• The lost profit due to a lost customer (cost of underage):
• Hence, intuition tells us that the newsboy should order
more than the average demand. But, how much more?
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Single Period Problem : Model
Inventory level
1) Order quantity
Q is determined
Q
2) Random
demand during
the period
?
0
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leftover
inventory
or, lost sales
3) Relevant costs
are realized at the
end of the period
4
Model Environment
• Relatively short selling season (weeks, 2 months,…) with a
well-defined beginning and end
• At the beginning of the period, a decision is made on how
much to order or produce (Q)
• The demand (D) is uncertain. Although we don’t know
exactly what value D is going to take, we have a forecast
on its distribution:
F(a)= P(D<a),
the cumulative distribution function (cdf) of D
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Model Environment
• When the total demand in the period exceeds the stock
available, there is an associated underage cost, cu
– cost per unit of unsatisfied demand
• When the total demand is less than the stock available,
overage cost is incurred, co
– cost per unit of positive inventory at the end of the period
• Objective: Minimize the total underage and overage cost
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Development of the Cost Function
• Define G(Q,D) as the total overage and underage cost
incurred at the end of the period when Q units are ordered and
the demand is realized as D
G (Q, D ) = co max{0, Q - D} + cu max{0, D - Q}
G (Q) = E[G (Q, D )]


0
Q
0

0
Q
= co  max{0, Q - x} f ( x)dx + cu  max{0, x - Q} f ( x)dx
= co  (Q - x) f ( x)dx + cu  ( x - Q ) f ( x) dx
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Leibniz’s Rule
• The optimal value of the order quantity Q?
– if necessary conditions are satisfied, find Q that satisfies
G(Q) = 0
• How to find the derivative of an integral?
b (Q )
d
g (Q, x)dx =

dQ a (Q )
b (Q )
d
d
dg( x, Q)
b(Q) g (Q, b(Q)) a(Q) g (Q, a(Q)) + 
dx
dQ
dQ
dQ
a (Q )
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Optimal Policy
Q

dG(Q)
= co  1 f ( x)dx + cu  (-1) f ( x)dx
dQ
0
Q
= co F (Q) - cu (1 - F (Q))
d 2G(Q)
= (co + cu ) f X (Q)  0
2
dQ
The function is convex
(the second order
condition is satisfied)
G' (Q* ) = (co + cu )F (Q* ) - cu = 0
cu
F (Q ) =
= T hecriticalratio
co + cu
*
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The CDF Plot
Cumulative probability
F(Q)
1
cu
cu + co
Q
Q*
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Normally Distributed Demand
X ~ N (,  2 )
E[ X ] = ,
Var(X) =  2
where (x) is the cumulative distribution function of
the standard normal random variable N(0,1)
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Normally Distributed Demand
T hen,theoptimalityequationcan be writtenas :
cu
Q- 

=
   cu + co
we get Q* =  + z
cu
where z statistic is the
th percentile of the standard
cu + co
normal random variable.
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The Standard Normal Distribution
Area =
cu
cu + co
 (z )
0
z
 =0
 =1
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Normally Distributed Demand: Insights
Q* =  + z
• Q* increases when
– mean demand increases
• What happens when variability increases?
– Q* increases if z > 0 (if cu/(cu+ co) > 0.5)
– Q* decreases if z < 0 (if cu/(cu+ co) < 0.5)
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Back to the Newsboy Example
cu
1
F(Q*)=
=
= 0.67
cu + co 1 + 0.5
Find Q* such that
•
•
Q* = μ + zσ
11.73+ (0.44*4.74) =
13.81 ~ order 14 items
0.08
0.07
Probability
z = NORMSINV(0.67)
= 0.439913
0.09
0.06
0.05
0.04
0.03
0.02
0.01
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
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Area=0.67
Sales
15
Another Newsboy Example
• Assume now that the newsboy cannot return unsold
copies to the publisher. In this case,
•
0.09
Q* = μ + zσ
0.08
Probability
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
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Area=0.25
Sales
16
Performance Measures
• Expected lost sales =
• where L(z) is the standard normal loss function (tabulated)
• Expected sales =
• where μ is the expected demand
• Exp. leftover inventory =
• Expected profit =
• Fill rate =
• the expected ratio of satisfied demand
• In-stock probability =
• the probability that all demand is satisfied (i.e., no stockout)
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Example with Uniform Demand
• Consider a case where demand is uniformly distributed
over [50,150]. cu=$10, co=$7. For uniform demand,
Solve
cu
Q - 50
10
F (Q) =
=
=
100
cu + co 17
10
 Q = 50 + 100  109
17
*
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Revenue Management Example
• Turkish Airlines has decided to offer a one-month advancepurchase discount ticket on its Istanbul-Munich route for $225
instead of the regular price of $325
• The flight has a capacity of 150 passengers
• Demand for regular tickets follow a normal distribution with
mean 60 and standard deviation of 15
– independent of the number of discount ticket customers
• Suppose that all discount tickets would be sold as soon as they
are released
• Determine the optimum number of seats that should be
reserved for the regular passengers
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Revenue Management Example
• Suppose we reserved X seats for the full-fare passengers,
and the number of full fare passengers that show up is D
• We will lose $100*(D-X) if D>X and
we will lose $225*(X-D) if X>D
• This problem can be modeled as a newsboy problem with
• The critical ratio is calculated as
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The Case of Discrete Demand
• Suppose we have an item for which cu / (cu+ co) = 0.68
Number of
units, Q
1
2
3
4
5
Total
P(D=Q)
0.2
0.2
0.3
0.2
0.1
1
P(D<=Q)
0.2
0.4
0.7
0.9
1.0
--
• Q* is the
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In Class Exercise
• Semicon uses nitric acid to produce semiconductors
• Nitric acid has a shelf-life of only three months
• Semicon’s need for nitric acid: Uniform[1000, 3000]
gallons for the coming three months
• Cost of acid: $150/gallon
• Acid storage cost: $35/gallon
– assume that this cost is incurred at the end of the 3-month period
• Leftover acid needs to be disposed of, costs $75/gal
• If the company runs out of acid during the 3-month period,
can place emergency order for $600/gal
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In Class Exercise:
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