Newsvendor model
The newsvendor (or newsboy or single-period[1] or salvageable) model is a mathematical model in operations management and applied economics used to determine optimal inventory levels.
It is (typically) characterized by fixed prices and uncertain demand for a perishable product.
This model is also known as the newsvendor problem or newsboy problem by analogy with the situation faced by a newspaper vendor who must decide how many copies of the day's paper to stock in the face of uncertain demand and knowing that unsold copies will be worthless at the end of the day.
The mathematical problem appears to date from 1888[2] where Edgeworth used the central limit theorem to determine the optimal cash reserves to satisfy random withdrawals from depositors.
[3] According to Chen, Cheng, Choi and Wang (2016), the term "newsboy" was first mentioned in an example of the Morse and Kimball (1951)'s book.
According to Evan Porteus, Matt Sobel coined the term "newsvendor problem".
[5] The modern formulation relates to a paper in Econometrica by Kenneth Arrow, T. Harris, and Jacob Marshak.
[6] More recent research on the classic newsvendor problem in particular focused on behavioral aspects: when trying to solve the problem in messy real-world contexts, to what extent do decision makers systematically vary from the optimum?
Experimental and empirical research has shown that decision makers tend to be biased towards ordering too close to the expected demand (pull-to-center effect[7]) and too close to the realisation from the previous period (demand chasing[8]).
This model can also be applied to period review systems.
representing demand, each unit is sold for price
The solution to the optimal stocking quantity of the newsvendor which maximizes expected profit is:
denotes the generalized inverse cumulative distribution function of
Intuitively, this ratio, referred to as the critical fractile, balances the cost of being understocked (a lost sale worth
The critical fractile formula is known as Littlewood's rule in the yield management literature.
In the following cases, assume that the retail price,
Therefore, the optimal inventory level is approximately 59 units.
In this situation, the optimal purchase quantity is zero to avoid a marginal loss.
To derive the critical fractile formula, start with
is monotone non-decreasing, this second derivative is always non-positive, so the critical point determined above is a global maximum.
If the demand D exceeds the provided quantity q, then an opportunity cost of
represents lost revenue not realized because of a shortage of inventory.
, then (because the items being sold are perishable), there is an overage cost of
This problem can also be posed as one of minimizing the expectation of the sum of the opportunity cost and the overage cost, keeping in mind that only one of these is ever incurred for any particular realization of
, is This is obviously the negative of the derivative arrived at above, and this is a minimization instead of a maximization formulation, so the critical point will be the same.
Assume that the 'newsvendor' is in fact a small company that wants to produce goods to an uncertain market.
captures the expected shortage quantity; its complement,
, denotes the expected product quantity in stock at the end of the period.
[10] On the basis of this cost function the determination of the optimal inventory level is a minimization problem.
So in the long run the amount of cost-optimal end-product can be calculated on the basis of the following relation:[1]
