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Newsvendor model

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Newsvendor model

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The newsvendor (or newsboy or single-period^[1]) model is
a mathematical model in operations management and applied
economics used to determine optimal inventorylevels.
It is (typically) characterized by fixed prices and uncertain demand for a perishable product. If the inventory level is $q$ ,
each unit of demand above $q$ is
lost. 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.

[hide]

[edit]Profit
function

The standard newsvendor profit function
is

$\pi =E\left[p\min (q,D)\right]-cq$

where $D$ is
a random variable with probability
distribution $F$ representing
demand, each unit is sold for price $p$ and
purchased for price $c$ ,
and $E$ is
the expectation operator. The solution to the optimal stocking quantity
of the newsvendor which maximizes expected profit is:

$q=F^{-1}\left( \frac{p-c}{p}\right)$

where $F^{-1}$ denotes
the inverse cumulative
distribution function of $D$ .

Intuitively, this ratio, referred to as the critical fractile, balances the cost of being understocked (a lost sale worth $(p-c)$ )
and the total costs of being either overstocked or understocked (where the cost of being overstocked is the inventory cost, or $c$ so
total cost is simply $p$ ).

[edit]Numerical
example

Assume that: retail price is $p=7$ [$/unit]
and purchase price is $c=5$ [$/unit].
Furthermore the $D$ demand
follows a uniform distribution (continuous) between $D_\min = 50$ and $D_\max = 80$ .

$q_\text{opt}=F^{-1}\left( \frac{7-5}{7}\right)=F^{-1}\left( 0.285 \right) = D_\min+(D_\max-D_\min) \cdot 0.285 = 58.55\approx59.$

Therefore optimal inventory level is approximately 59 units.

[edit]Extreme
situation

If $p<c$ (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.

[edit]Cost
based optimization of inventory level

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:

$K(q) = c_f + c_v (q-x) + p E\left[\max(D-q,0)\right] + h E\left[\max(q-D,0)\right]$

where the individual parameters are the following:

$c_f$ –
fixed cost. This cost always exists when the production of a series is started. [$/production]
$c_v$ –
variable cost. This cost type expresses the production cost of one product. [$/product]
$q$ –
The product quantity in the inventory. The decision of the inventory control policy concerns the product quantity in the inventory after the product decision. This parameter includes the initial inventory as well. If nothing is produced, then this quantity
is equal to the initial quantity, i.e. concerning the existing inventory.
$x$ –
Initial inventory level. We assume that the supplier possesses $x$ products
in the inventory at the beginning of the demand of the delivery period.
$p$ –
penalty cost (or back order cost). If there is less raw material in the inventory than needed to satisfy the demands, this is the penalty cost of the unsatisfied orders. [$/product]
$E[D]$ –
Expected value of the $D$ stochastic
variable.
$D$ –
This means the demand from the receiver for the product, which is an optional probability variable. [unit]
$h$ –
inventory and stock holding cost. [$ / product]

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:^[1]

$q_\text{opt} = F^{-1}\left( \frac{p-c_v}{p+h}\right)$

[edit]See
also

[edit]References

^ ^a ^b William
J. Stevenson, Operations Management. 10th edition, 2009; page 581

[edit]Further
reading

Ayhan, Hayriye, Dai, Jim, Foley, R. D., Wu, Joe, 2004: Newsvendor Notes, ISyE 3232 Stochastic Manufacturing & Service Systems. [1]

【上篇】程序分析的方法
【下篇】数据中心发展趋势：绿色+云计算

作者: zooming

该日志由 zooming 于5年前发表在综合分类下，最后更新于 2019年05月02日.
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