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Newsvendor Model OIS 5620
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A decision tool is needed to make the best out of this difficult decision where you try to match supply with demand. You try to order the amount you think you can sell. What is the tool called?
The newsvendor model
is also the value that minimizes the expected absolute difference between the actual demand and the production quantity.
the forecast
considers a setting in which you have only one production or procurement opportunity. Because that opportunity occurs well in advance of a single selling season, you receive your entire order just before the selling season starts.
Newsvendor model
In the newsvendor model, the Cost should only include costs that depend on the number of units ordered. Fixed costs should not be included. Just the variable unit cost?
True
value of an item that is left over at the end of the season
Salvage value
It's possible to have a negative salvage value?
True for example if something has chemicals in it and costs money to dispose of then it has a negative salvage value
represents a situation in which a decision maker must make a single bet (order quantity) before some random event occurs (demand). There are costs if bet turns out to high and costs if bet turns out too low.
Newsvendor Model
The newsvendor model's objective is?
to bet an amount that correctly balances those opposing forces of supply and demand.
To implement the newsvendor model we need what 2 things?
our cost and how much demand uncertainty we have
The newsvcndor model balances the cost of ordering too much against the cost of ordering
too little. To do this, we need to understand how much demand unce1tainty there is for the
Hammer 312, which essen tially means we need to be able to answer the following question:
What is the probability demand will be less than or equal to Q units?
Recall from
statistics, every random variable is defined by its distribution function,
F(Q)
What is F(Q)
is the
probability the outcome of the random variable is Q or lower. In this case the random variable is demand
we refer to the distribution function, F(Q). as our
demand forecast because it gives us a complete picture of the demand uncertainty we face.
Distribution functions come in what two forms
1. Discrete distribution functions
2. Poisson distributions
Explain the discrete distribution function
can be defined in the form of a table: There is a set of possible outcomes and each possible outcome has a probability associated with it.
Give example of simple discrete distribution function
Q F(Q)
2700 .25
3200 .75
4200 1.00
is an example of a discrete distribution function that we will use continuously
Poisson distribution
With ___ ____ functions there are an unlimited number of possible outcomes
continuous distributions
Both the exponential and the normal are continuous distribution functions?
True
The normal distribution is defined by two parameters:
its mean and its standard deviation
We use ___ to represent the mean of the distribution and __ to represent the standard deviation.
Mu (looks like a u); sigma (looks like Q)
What is A/F Ratio
Ratio of actual demand to forecast
An accurate forecast has an A/F ratio =
1
while an AIF ratio above I indicates the forecast
was too
low
AIF ratio below 1 indicates the forecast was too
high
AF ratio X Forecast =
Actual Demand
Expected actual demand =
Expected A/F Ratio * Forecast
Standard deviation of (actual) demand =
Standard deviattion of A/F ratios X Forecast
Expected actual demand, or expected demand for short, is what we should choose
for the _____ for our normal distribution.
mean
How do you find expected A/F Ratio?
Average A/F ratio
is a particular normal distribution: its mean is 0 and its standard
deviation is I
Standard Normal
How do you find Z
(Q-u)/Standard Deviation
Expected Demand =
Expected A/F ratio X Forecast
The expected gain from the first unit, which
equals the probability of
selling the first unit times the gain from the first unit.
is the per-unit cost of overordering
overage cost
is the per-unit opportunity cost of
underordering
underage cost
Explain expected gain and expected loss
The expected gain of ordering one is 100% 80 because it for sure will sell and 100% 0 expected loss because it will sell. Where gain and loss meat is how much you should order. See 252
the overage and underage costs are
defined for a single unit?
True
The expected loss on a unit
is the cost of having the unit in inventory (the overage cost)
times the probability it is left in inventory (F(Q))
The
expected gain on a unit is
the benefit of selling a unit (the underage cost) times the probability
the unit is sold, which in this case occurs if demand is greater than Q.
The probability
demand is greater than Q is
(I - F(Q)).
Therefore, the expected gain is
Cu * (I - F(Q))
Profit maximizing order quantity formula?
F(Q) = Cu/(Co + Cu)
ratio with the
underage and overage costs is called the
critical ratio
critical ratio?
Cu/(Cu + Co)
Is the probability demand is Q or lower
F(Q)
To convert our z into an order quantity that makes sense for our actual demand
forecast, we use the following equation:
Q = u + (z*std dev). Where u is the expected actual demand
Round-up rule
Whenever you are looking up a target value in a table and the target
value falls between two entries, choose the entry that leads to the
larger order quantity.
Steps to find the order quantity that maximizes expected profit in newsvendor model
First find critical ratio.
Find the z value using critical ratio
Convert the Z into order quantity doing u + z*standard deviation
which is the expected number of units by which
demand (a random variable) exceeds the order quantity (a fixed threshold)
expected lost sales
Expected lost sales =
Standard deviation * L(z)
What is L(z)
loss function with the standard normal distribution
How do you find L(z)
Find the z value and use the Loss function table to find value.
Expected Sales + Expected lost sales =
expected demand
= u - Expected lost sales
expected sales
expected sales is always more than expected demand (because expected lost
sales is never negative)?
False. It's always less
Steps to finding expected lost sales?
Find the z value
Use z to find L(z) use lost table
Then use expected lost sales formula, which is standard deviation * L(z)
is the average amount that demand (a random vmiable) is less
than the order quantity (a fixed threshold)
expected leftover inventory
is the average
amount by which demand exceeds the order quantity.)
Expected lost sales
Q (ordered) - Expected sales =
Expected leftover inventory
We earn Price- Cost on each unit sold and we lose Cost - Salvage value on each unit we
do not sell, so our expected profit is
Expected profit =
[( Price - Cost) X Expected sales)
- [(Cost - Salvage value) X Expected leftover inventory]
is the probability the firm ends the season having satisfied all demand.
In-stock probability
In-stock probability =
F(Q)
is the probability the firm stocks out for some customer during the
selling season (i.e .. a lost sale occurs).
The stockout probability
Stockout probability =
I - F(Q)
is the probability a c ustomer is able to purchase a unit
(i.e .• does not experience a stockout).
Fill Rate
Interestingly, this is not the same as the in-stock
probability, which is the probability that all demand is satisfied?
True
Steps to find In stock and stockout probability
Find z statistics
Find z value on table.
Now you have In stock probability. Do 1 - amount to get stockout probability
Choose the Lower z statistic?
False choose the higher one
Now that we have detailed the process of implementing the newsvendor model, it is worthwhile
to step back and consider the managerial lessons it implies.
With respect to the forecasting process, there are three key lessons.
• For each product, it is insufficient to have just a forecast of expected demand. We
also need a fore.cast for how variable demand will be about the forecast. That uncertainty
in the forecast is captured by the standard deviation of demand.
• It is important to track actual demand.
• You need to keep track of past forecasts and forecast errors in order to assess the
standard deviation of demand.
The profit-maximizing order quantity generally does not equal expected demand. If
the underage cost is greater than the overage cost (i.e., it is more expensive to lose a sale
than it is to have leftover inventory), then the profit-maximizing order quantity is ____ than expected demand
larger
Important lessons from the order quantity choices.
• The profit-maximizing order quantity generally does not equal expected demand.
• The order quantity decision should be separated from the forecasting process.
• Explicit costs should not be overemphasized relative to opportunity costs.
• It is important to recognize that choosing an order quantity to maximize expected
profit is only one possible objective.
Know all formulas on 262
.
Why has Supply Chain Management
become so important?
Higher competitiveness (often from foreign companies) in the marketplace allows for
less and less freedom on the revenue side.
Uncertainty in product demand is increasing:
Importance of incentives in supply chains is growing:
Increased availability of data due to computerization.
Better availability of computing power to execute complex optimization algorithms.
What are the 4 newsvendor model implementation steps
1. Gather economic inputs like selling price, production cost, salvage value of inventory
2. Generate a demand model by determining a distribution function that accurately reflects the possible demand outcomes
3. Choose an objective
4. Choose a quantity to order
Steps in making the empirical distribution function of forecast accuracy
sort the N observations from lowest A/F ratio to highest A/F ratio
Then find the percentile, which is the Rank/N. Rank for lowest A/F ratio is 1. Rank for 2nd lowest is 2 etc
How to find order quantity given A/F ratio
Forecast * A/F
How to convert between normal distributions
Subtract the mean to make zero. If standard deviation is 25 then 125 would be standard deviation of 1. 150 is 2 and so on. See slide on page 13
Computing the critical ratio for poisson demand distribution is different than for normal distribution?
False. It's the same
What's the difference between finding the optimal quantity on normal vs poisson?
Use the poisson table on poisson
Steps to find expected lost sales
1. Find z value
2. Use standard normal Loss function table to find L(0.26)
3. Evaluate lost sales for the actual normal distribution. Expected lost sales = standard dev * L(0.26)
Suppose we wish to find the order quantity for the Hammer 3/2 that
minimizes left over inventory while generating at least a 99% in-stock
probability. What are the steps?
1. find z statistic that is 99%. Choose higher of two.
2. Convert z statistic to actual quantity demand
Suppose we wish to find the order quantity for the Hammer 3/2 that
minimizes left over inventory while generating at least a 99% fill rate. What are the steps?
1. Find the lost sales with standard normal distribution that yields target fill rate. Use L(z) = (u/q)(1 - Fill rate)
2. Find the z-statistic that yeilds the lost sales found in step 1
3. Convert z statistic into an order quantity for actual demand.
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