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Newsboy  One Time Decisions
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by
TweetLuis Jesús Pérez Rivera
on 20 November 2014Transcript of Newsboy  One Time Decisions
Newsboy Model
(One Time Decisions)
M Sc. Luis Jesús Pérez Rivera
MSc.ljperez@gmail.com
Newsboy Model
Scenario in which you have
only one production / procurement opportunity.
Opportunity occurs well in advance of a single selling season.
Stochastic
Demand (random variable)
occurs during the selling season.
if
D>Q
: unsatisfied demand and thus an opportunity cost.
# Units short = DQ
if
D<Q
: leftover inventory. There may be a salvage value.
# Exceeding units = QD
Then the
expected
total cost of ordering
Q
is:
Expected Values (Discrete probability functions)
Let be any given function of , then
Let
X be the result of rolling a fair dice one time
.
You will win
h(
X
)=(1/
X
)*100
dollars. What is the expected value of the prize?
example
example
Expected Values (Continuous probability functions)
Let be any given function of , then
Overage
Shortage
Then the
expected
total cost of ordering
Q
is:
To get the minimum expected cost, we aim for:
which with some manipulation yields:
exactly the same than:
Isn't it true that .... ?
If so, then:
therefore:
Recall that is the
inverse
cumulative distribution function.
This relation is also known as "critical ratio"
=
purchase or production cost
+
Any other expenses caused by shortages
=
contribution margin or profit
Any other expenses caused by exceeding units
+

Salvage value
Newsboy in 6 easy steps
(Continuous distributions 
not Normal
)
1. Estimate , and .
2. Get (if not given) and .
3. Estimate
4. Calculate the expected exceeding units
5. Calculate the expected short units
6. Calculate the expected total cost.
Newsboy in 6 easy steps
(Normal Distribution)
1. Estimate
,
and
.
2. Estimate
3. Calculate
4. Calculate the expected units short:
5. Calculate the expected exceeding units:
6. Calculate the expected total cost.
Expected profit
Expected sales = E(Demand)  E(units short)
Expected profit = (pricecost)*Expected sales  Co*E(exceeding units)
There isn't a standard and definitive idea about how to calculate the expected profit in the newsboy model scenario, but we'll consider:
It could be negative. Why?
If you don't like using the z and L(z) tables in paper consider using this functions in Excel:
for the normal standard inverse cumulative distribution function:
= INV.NORM.ESTAND(
z
)
for the normal standard loss function:
=DISTR.NORM.ESTAND.N(
z
,0)
(
z
*(1DISTR.NORM.ESTAND.N(
z
,1)))
Discrete distributions
Poisson distribution
You can say that a random variable (eg. X) has a poisson distribution if the mass probability function of X is:
for
lambda > 0
where lambda is the expected value (mean). In our case the random variable is the demand, so:
Continuous distributions
(eg. exponential)
(eg. Poisson)
possible values for a random variable can be listed
probability mass function
cumulative probability is done with a sum
density function
cumulative probability is done with an integral
Full transcript(One Time Decisions)
M Sc. Luis Jesús Pérez Rivera
MSc.ljperez@gmail.com
Newsboy Model
Scenario in which you have
only one production / procurement opportunity.
Opportunity occurs well in advance of a single selling season.
Stochastic
Demand (random variable)
occurs during the selling season.
if
D>Q
: unsatisfied demand and thus an opportunity cost.
# Units short = DQ
if
D<Q
: leftover inventory. There may be a salvage value.
# Exceeding units = QD
Then the
expected
total cost of ordering
Q
is:
Expected Values (Discrete probability functions)
Let be any given function of , then
Let
X be the result of rolling a fair dice one time
.
You will win
h(
X
)=(1/
X
)*100
dollars. What is the expected value of the prize?
example
example
Expected Values (Continuous probability functions)
Let be any given function of , then
Overage
Shortage
Then the
expected
total cost of ordering
Q
is:
To get the minimum expected cost, we aim for:
which with some manipulation yields:
exactly the same than:
Isn't it true that .... ?
If so, then:
therefore:
Recall that is the
inverse
cumulative distribution function.
This relation is also known as "critical ratio"
=
purchase or production cost
+
Any other expenses caused by shortages
=
contribution margin or profit
Any other expenses caused by exceeding units
+

Salvage value
Newsboy in 6 easy steps
(Continuous distributions 
not Normal
)
1. Estimate , and .
2. Get (if not given) and .
3. Estimate
4. Calculate the expected exceeding units
5. Calculate the expected short units
6. Calculate the expected total cost.
Newsboy in 6 easy steps
(Normal Distribution)
1. Estimate
,
and
.
2. Estimate
3. Calculate
4. Calculate the expected units short:
5. Calculate the expected exceeding units:
6. Calculate the expected total cost.
Expected profit
Expected sales = E(Demand)  E(units short)
Expected profit = (pricecost)*Expected sales  Co*E(exceeding units)
There isn't a standard and definitive idea about how to calculate the expected profit in the newsboy model scenario, but we'll consider:
It could be negative. Why?
If you don't like using the z and L(z) tables in paper consider using this functions in Excel:
for the normal standard inverse cumulative distribution function:
= INV.NORM.ESTAND(
z
)
for the normal standard loss function:
=DISTR.NORM.ESTAND.N(
z
,0)
(
z
*(1DISTR.NORM.ESTAND.N(
z
,1)))
Discrete distributions
Poisson distribution
You can say that a random variable (eg. X) has a poisson distribution if the mass probability function of X is:
for
lambda > 0
where lambda is the expected value (mean). In our case the random variable is the demand, so:
Continuous distributions
(eg. exponential)
(eg. Poisson)
possible values for a random variable can be listed
probability mass function
cumulative probability is done with a sum
density function
cumulative probability is done with an integral