# Newsvendor problem

We consider a decision situation that arises quite often in real life and that is known in the literature as "Newsvendor problem", "Newsboy problem", or "Christmas Tree problem". The following situation is considered. In the immediately forthcoming period a random demand is expected. At the beginning of the period we have to decide about the placement of a single replenishment order from which the demand is filled. As there is only a single opportunity to place the order, the inventory on hand after the order arrival must cover the demand that occurs until the end of the period. The replenishment lead time is assumed to be zero.

Let the inventory on hand before the ordering decision be equal to $b$. It is often assumed that $b=0$. The inventory on hand immediately after the ordering decision is $S$. Thus, the order quantity is $(S-b)$. If $S$ is not sufficient to fill the complete demand arriving in the upcoming period, then shortage costs of $c_f$ per unit short arise. In the case that the demand is smaller than $S$, a certain amount of inventory is left over at the end of the period and for each unit unsold the cost is $c_o$. These costs may be holding costs for the storage of the product until the time of the next decision situation of this kind, or costs for the disposal of the product.

We consider first the simple case that there are no fixed ordering costs. After that we include these into the analysis.

### Single period model without fixed costs

Suppose that ordering is possible without incurring fixed ordering costs and that the inventory on hand just before the ordering decision is $b=0$. We search for the optimal order level $S$ (i.e. the inventory on hand after the ordering decision). As $b=0$ the order quantity is $q=S$.

Let $X$ denote the random demand in the upcoming period. At the end of the period, after the realization of the demand, two situations with the following costs are then possible:

 Demand Situation Costs Low Inventory on Hand $=S-X$ $c_o\cdot (S-X)$ High Shortage $=X-S$ $c_f\cdot (X-S)$

The expected costs at the end of the period amount to

$C(S) = c_o\cdot \displaystyle{\int_{0}^S} \underbrace{(S-x)\cdot f(x)\cdot dx}_\mathrm{overstock} + c_f\cdot \displaystyle{\int_{S}^\infty} \underbrace{(x-S)\cdot f(x)\cdot dx}_\mathrm{shortage}$

or

$C(S) = c_o\cdot \displaystyle{\int_{0}^S} (S-x)\cdot f(x)\cdot dx +\; c_f\cdot \left[ \displaystyle{\int_{0}^\infty} (x-S)\cdot f(x)\cdot dx - \displaystyle{\int_{0}^{S}} (x-S)\cdot f(x)\cdot dx \right]$

or

$C(S) = (c_o+c_f)\cdot \displaystyle{\int_{0}^S} (S-x)\cdot f(x)\cdot dx + c_f\cdot \displaystyle{\int_{0}^\infty} (x-S)\cdot f(x)\cdot dx$

or

$C(S) = (c_o+c_f)\cdot \displaystyle{\int_{0}^S} (S-x)\cdot f(x)\cdot dx + c_f\cdot ( E\{X\}- S)\qquad$(1)

In order to determine the optimal value of $S$, the first derivative of (1) with respect to $S$ is taken and set to zero:

$\frac{dC(S)}{dS} = (c_o+c_f) \cdot \displaystyle{\int_{0}^S} f(x)\cdot dx - c_f \stackrel{!}{=} 0\qquad$(2)

The second derivative with respect to $S$ is

$\frac{d^2C(S)}{dS^2} = (c_o+c_f) \cdot f(S) > 0$

This function is always $>0$ as $f(x)$ cannot become negative. Therefore, the cost function $C(S)$ is convex.

From (2) we can derive the following optimality condition ("critical ratio"):

$\displaystyle{\int_{0}^{S_{\mathrm{opt}}}} f(x)\cdot dx =F(S_{\mathrm{opt}}) = \frac{c_f}{c_o+c_f}$

This is illustrated with an example. The period demand is normally distributed with mean $\mu_X=100$ and standard deviation $\sigma_X=30$. The shortage costs are $c_f=30$ and the overstocking costs are $c_o=10$. We obtain the following results:

$S_{\mathrm{opt}} = \big[ S \mid F(S) = \frac{30}{10+30}=0.75 \big]$

With $v=\frac{S-\mu}{\sigma}$ we can write:

$v_{\mathrm{opt}} = \big[ v \mid \Phi_N(v) = 0.75 \big] = 0.6743$

and

$S_{\mathrm{opt}} = 100 + 0.6743\cdot 30 \simeq 120.23$