10 terms

Ch.12 The Newsvendor Model

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Newsvendor model summary
●The model can be applied to settings in which ...
-There is a single order/production/replenishment opportunity.
-Demand is uncertain.
-There is a "too much-too little" challenge:
●If demand exceeds the order quantity, sales are lost.
●If demand is less than the order quantity, there is left over inventory.

●Firm must have a demand model that includes an expected demand and uncertainty in that demand.
-With the normal distribution, uncertainty in demand is captured with the standard deviation parameter.

●At the order quantity that maximizes expected profit the probability that demand is less than the order quantity equals the critical ratio:
-The expected profit maximizing order quantity balances the "too much-too little" costs.
Newsvendor problem
-Relatively short selling season / life cycle
-Advance commitment to stock
-Considerable forecast uncertainty
-Main cost elements:
●Shortage ('underage') costs, cu
●Excess ('overage') costs, co
Newsvendor model implementation steps
●Generate a demand model:
-Determine a distribution function that accurately reflects the possible demand outcomes, such as a normal distribution function.

●Gather economic inputs:
-Selling price, production/procurement cost, salvage value of inventory

●Choose an objective:
-e.g. maximize expected profit or satisfy an in-stock probability.

●Choose a quantity to order.
What's a demand model?
●A demand model specifies what demand outcomes are possible and the probability of these outcomes.
●Traditional distributions from statistics can be used as demand models:
-e.g., the normal, gamma, Poisson distributions
Newsvendor model performance measures
●For any order quantity we would like to evaluate the following performance measures:
-In-stock probability
●Probability all demand is satisfied
-Stockout probability
●Probability some demand is lost
-Expected lost sales
●The expected number of units by which demand will exceed
the order quantity
-Expected sales
●The expected number of units sold.
-Expected left over inventory
●The expected number of units left over after demand (but before salvaging)
-Expected profit
In stock probability
●The in-stock probability is the probability all demand is satisfied.

●All demand is satisfied if demand is the order quantity, Q, or smaller.
-If Q = 3000, then to satisfy all demand, demand must be 3000 or fewer.

●The distribution function tells us the probability demand is Q or smaller!

●Hence, the In-stock probability = F(Q) = F(z)
What is a distribution function?
●The distribution function tells you the probability the outcome will be a particular value or smaller.
-e.g. the probability demand will be 5 or fewer units.

●We will most often work with distribution functions to describe a demand model.

●Distribution functions always start low and increase towards 1.0 (100%)
for high values of demand.
Overage and underage cost
●Co = overage cost
-The consequence of ordering one more unit than what you would have ordered had you known demand.
●Suppose you had left over inventory (you over ordered). Co is the increase in profit you would have enjoyed had you ordered one fewer unit.
-For the Hammer 3/2 Co = Cost - Salvage value = c - v = 110 - 90 = 20

●Cu = underage cost
-The consequence of ordering one fewer unit than what you would have ordered had you known demand.
●Suppose you had lost sales (you under ordered). Cu is the increase in profit you would have enjoyed had you ordered one more unit.
-For the Hammer 3/2 Cu = Price - Cost = p - c = 190 - 110 = 80
Other measures of service performance
●The stockout probability is the probability some demand is not satisfied:
-Some demand is not satisfied if demand exceeds the order quantity, thus...
-Stockout probability = 1 - F(Q)
= 1 - In-stock probability
= 1 -0.4364 = 56.36%

●The fill rate is the fraction of demand that can purchase a unit:
-The fill rate is also the probability a randomly chosen customer can purchase a unit.
-The fill rate is not the same as the in-stock probability!
●e.g. if 99% of demand is satisfied (the fill rate) then the probability all demand is satisfied (the in-stock) need not be 99%
What's the Poison distribution and what is it good for?
●Defined only by its mean (standard deviation = square root(mean))
●Does not always have a "bell" shape, especially for low demand.
●Discrete distribution function: only non-negative integers
●Good for modeling demands with low means (e.g., less than 20)
●If the inter-arrival times of customers are exponentially distributed, then the number of customers that arrive in a given interval of time has a Poisson distribution.
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