Ch.12 The Newsvendor Model

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

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

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.

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

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 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.

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)

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.

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%