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The newsvendor (or newsboy or single-period or perishable ) 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. If the inventory level is , each unit of demand above is lost in potential sales. 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 where Edgeworth used the central limit theorem to determine the optimal cash reserves to satisfy random withdrawals from depositors. The modern formulation dates from a 1951 paper in Econometrica by Kenneth Arrow, T. Harris, and Jacob Marshak.
The standard newsvendor profit function is
where is a random variable with probability distribution representing demand, each unit is sold for price and purchased for price , is the number of units stocked, and is the expectation operator. The solution to the optimal stocking quantity of the newsvendor which maximizes expected profit is:
|Critical fractile formula|
Intuitively, this ratio, referred to as the critical fractile, balances the cost of being understocked (a lost sale worth ) and the total costs of being either overstocked or understocked (where the cost of being overstocked is the inventory cost, or so total cost is simply ).
Assume that: retail price is [$/unit] and purchase price is [$/unit]. Furthermore the demand follows a uniform distribution (continuous) between and .
Therefore optimal inventory level is approximately 59 units.
Therefore optimal inventory level is approximately 39 units.
q opt = Mu + sigma x zinv x (2/7) Go to MSExcel. NORMSINV (0.285714) = - 0.56595 Therefore q= 50 + 20 (-0.56595) = 38.69 units
Therefore optimal inventory level is approximately 45 units.
If (i.e. the retail price is less than the purchase price), the numerator becomes negative. In this situation, it isn't worth keeping any item in the inventory.
Assuming that the 'newsvendor' is in fact a small company who wants to produce goods to an uncertain market. In this more general situation the cost function of the newsvendor (company) can be formulated in the following manner:
where the individual parameters are the following:
On the basis of the 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: