“Newsvendor model”的意思、由来-开放百科全书

The newsvendor (or newsboy or single-period^[1] 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.

History

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/^[4] The modern formulation relates to a paper in Econometrica by Kenneth Arrow, T. Harris, and Jacob Marshak.^[5]

Profit function and the critical fractile formula

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:

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

The critical fractile formula is known as Littlewood's rule in the yield management literature.

Numerical examples

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

Uniform distribution

Normal distribution

Let demand,

, follow a normal distribution with a mean,

, demand of 50 and a standard deviation,

, of 20.

Lognormal distribution

Let demand,

, follow a lognormal distribution with a mean demand of 50,

, and a standard deviation,

, of 0.2.

Extreme situation

(i.e. the retail price is less than the purchase price), the numerator becomes negative. In this situation, it isn't worth keeping any items in the inventory.

Derivation of optimal inventory level

To derive the critical fractile formula, start with

and condition on the event

Since

is monotone non-decreasing, this second derivative is always non-positive, so the critical point determined above is a global maximum.

Alternative formulation

The problem above is cast as one of maximizing profit, although it can be cast slightly differently, with the same result. 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. On the other hand, if

, 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

. The derivation of this is as follows:

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.

Cost based optimization of inventory level

Assume that the 'newsvendor' is in fact a small company that 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:

, the first order loss function

captures the expected shortage quantity; its complement,

, denotes the expected product quantity in stock at the end of the period.^[6]

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]

Data-driven models

There are several data-driven models for the newsvendor problem. Among them, a deep learning model provides quite stable results in any kind of non-noisy or volatile data.^[7] More details can be found in a [https://oroojlooy.github.io/blog/newsvendor/ blog] explained the model^[8].

See also

References

1. ^¹William J. Stevenson, Operations Management. 10th edition, 2009; page 581
2. ^{{cite journal | author = F. Y. Edgeworth | authorlink = Francis Ysidro Edgeworth | year = 1888 | title = The Mathematical Theory of Banking | journal = Journal of the Royal Statistical Society | volume = 51 | issue = 1 | pages = 113–127 | url = | format = | accessdate = | jstor = 2979084}}
3. ^{{cite web|url=http://www.columbia.edu/~gmg2/4000/pdf/lect_07.pdf|title=IEOR 4000 Production Management Lecture 7|author=Guillermo Gallego|publisher=Columbia University|date=18 Jan 2005|accessdate=30 May 2012}}
4. ^{{cite journal |author1=R. R. Chen |author2=T.C.E. Cheng |author3=T.M. Choi |author4=Y. Wang | title = Novel Advances in Applications of the Newsvendor Model| journal = Decision Sciences | volume = 47 | pages = 8–10 | year = 2016 }}
5. ^K. J. Arrow, T. Harris, Jacob Marshak, Optimal Inventory Policy, Econometrica 1951
6. ^{{cite book | last = Axsäter | first = Sven | authorlink = Sven Axsäter | year = 2015 | title = Inventory Control | edition=3rd | publisher=Springer International Publishing | isbn = 978-3-319-15729-0 }}
7. ^{{cite arxiv|last=Oroojlooyjadid|first=Afshin|last2=Snyder|first2=Lawrence|last3=Takáč|first3=Martin|date=2016-07-07|title=Applying Deep Learning to the Newsvendor Problem|eprint=1607.02177|class=cs.LG}}
8. ^{{Cite web|url=https://oroojlooy.github.io/blog/newsvendor|title=Deep Learning for Newsvendor Problem|last=Afshin|date=2017-04-11|website=Afshin|language=en|access-date=2019-03-10}}