Newsvendor Model OIS 5620

We use ___ to represent the mean of the distribution and __ to represent the standard deviation.

Mu (looks like a u); sigma (looks like Q)

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.

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)

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

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

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

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

is an example of a discrete distribution function that we will use continuously

Poisson distribution

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

Give example of simple discrete distribution function

Q F(Q) 2700 .25 3200 .75 4200 1.00

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.

Expected Sales + Expected lost sales =

expected demand

is the average amount that demand (a random vmiable) is less than the order quantity (a fixed threshold)

expected leftover inventory

which is the expected number of units by which demand (a random variable) exceeds the order quantity (a fixed threshold)

expected lost sales

= u - Expected lost sales

expected sales

AIF ratio below 1 indicates the forecast was too

high

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

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 normal distribution is defined by two parameters:

its mean and its standard deviation

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

What is L(z)

loss function with the standard normal distribution

while an AIF ratio above I indicates the forecast was too

low

Expected actual demand, or expected demand for short, is what we should choose for the _____ for our normal distribution.

mean

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.

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.

Q (ordered) - Expected sales =

Expected leftover inventory

is the average amount by which demand exceeds the order quantity.)

Expected lost sales

In-stock probability =

F(Q)

Is the probability demand is Q or lower

F(Q)

Recall from statistics, every random variable is defined by its distribution function,

F(Q)

Profit maximizing order quantity formula?

F(Q) = Cu/(Co + Cu)

Choose the Lower z statistic?

False choose the higher one

expected sales is always more than expected demand (because expected lost sales is never negative)?

False. It's always less

Computing the critical ratio for poisson demand distribution is different than for normal distribution?

False. It's the same

is the probability a c ustomer is able to purchase a unit (i.e .• does not experience a stockout).

Fill Rate

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)

How do you find L(z)

Find the z value and use the Loss function table to find value.

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

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

How to find order quantity given A/F ratio

Forecast * A/F

Stockout probability =

I - F(Q)

is the probability the firm ends the season having satisfied all demand.

In-stock probability

Both the exponential and the normal are continuous distribution functions?

True

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

Interestingly, this is not the same as the in-stock probability, which is the probability that all demand is satisfied?

True

the overage and underage costs are defined for a single unit?

True

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

What's the difference between finding the optimal quantity on normal vs poisson?

Use the poisson table on poisson

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?

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.

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]

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.

With ___ ____ functions there are an unlimited number of possible outcomes

continuous distributions

ratio with the underage and overage costs is called the

critical ratio

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.

The probability demand is greater than Q is

(I - F(Q)).

How do you find Z

(Q-u)/Standard Deviation

Know all formulas on 262

.

An accurate forecast has an A/F ratio =

1

Distribution functions come in what two forms

1. Discrete distribution functions 2. Poisson distributions

AF ratio X Forecast =

Actual Demand

How do you find expected A/F Ratio?

Average A/F ratio

Therefore, the expected gain is

Cu * (I - F(Q))

critical ratio?

Cu/(Cu + Co)

Expected actual demand =

Expected A/F Ratio * Forecast

Expected Demand =

Expected A/F ratio X Forecast

What is A/F Ratio

Ratio of actual demand to forecast

value of an item that is left over at the end of the season

Salvage value

is a particular normal distribution: its mean is 0 and its standard deviation is I

Standard Normal

Expected lost sales =

Standard deviation * L(z)

Standard deviation of (actual) demand =

Standard deviattion of A/F ratios X Forecast

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

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

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 the probability the firm stocks out for some customer during the selling season (i.e .. a lost sale occurs).

The stockout probability

To implement the newsvendor model we need what 2 things?

our cost and how much demand uncertainty we have

is the per-unit cost of overordering

overage cost

The expected gain from the first unit, which equals the probability of

selling the first unit times the gain from the first unit.

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

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.

is also the value that minimizes the expected absolute difference between the actual demand and the production quantity.

the forecast

The newsvendor model's objective is?

to bet an amount that correctly balances those opposing forces of supply and demand.

is the per-unit opportunity cost of underordering

underage cost