Қор жағдайларын анықтауға арналған математикалық модельдер

Mathematical model to assist inventory levels
The newsvendor (or newsboy or single period 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. 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. 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. The problem was termed the "Christmas tree problem" and "newboy problem" in the 1960s and 1970s, and beginning in the 1980s gender neutral vocabulary like "newsperson" began to be used. According to Evan Porteus, Matt Sobel coined the term "newsvendor problem". The modern formulation relates to a paper in Econometrica by Kenneth Arrow, T. Harris, and Jacob Marshak. 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) and too close to the realisation from the previous period (demand chasing).

This model can also be applied to period review systems.

In the following cases, assume that the retail price, , is $7 per unit and the purchase price is , is $5 per unit. This gives a critical fractile of

Let demand, , follow a uniform distribution (continuous) between and

Therefore, the optimal inventory level is approximately 59 units.

Let demand, , follow a normal distribution with a mean, , demand of 50 and a standard deviation, , of 20. Therefore, optimal inventory level is approximately 39 units.

Let demand, , follow a lognormal distribution with a mean demand of 50, , and a standard deviation, , of 0.2. 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, the optimal purchase quantity is zero to avoid a marginal loss.