Numerical examples

In the following cases, assume that the retail price, ${\displaystyle p}$, is $7 per unit and the purchase price is ${\displaystyle c}$, is $5 per unit. This gives a critical fractile of ${\displaystyle {\frac {p-c}{p}}={\frac {7-5}{7}}={\frac {2}{7}}}$

Critical fractile formula

To derive the critical fractile formula, start with ${\displaystyle \operatorname {E} \left[{\min\{q,D\}}\right]}$ and condition on the event ${\displaystyle D\leq q}$:

${\displaystyle {\begin{aligned}&\operatorname {E} [\min\{q,D\}]=\operatorname {E} [\min\{q,D\}\mid D\leq q]\operatorname {P} (D\leq q)+\operatorname {E} [\min\{q,D\}\mid D>q]\operatorname {P} (D>q)\\[6pt]={}&\operatorname {E} [D\mid D\leq q]F(q)+\operatorname {E} [q\mid D>q][1-F(q)]=\operatorname {E} [D\mid D\leq q]F(q)+q[1-F(q)]\end{aligned}}}$

Now use

${\displaystyle \operatorname {E} [D\mid D\leq q]={\frac {\int \limits _{x\leq q}xf(x)\,dx}{\int \limits _{x\leq q}f(x)\,dx}},}$

where ${\displaystyle f(x)=F'(x)}$. The denominator of this expression is ${\displaystyle F(q)}$, so now we can write:

${\displaystyle \operatorname {E} [\min\{q,D\}]=\int \limits _{x\leq q}xf(x)\,dx+q[1-F(q)]}$

So ${\displaystyle \operatorname {E} [{\text{profit}}]=p\int \limits _{x\leq q}xf(x)\,dx+pq[1-F(q)]-cq}$

Take the derivative with respect to ${\displaystyle q}$:

${\displaystyle {\frac {\partial }{\partial q}}\operatorname {E} [{\text{profit}}]=pqf(q)+pq(-F'(q))+p[1-F(q)]-c=p[1-F(q)]-c}$

Now optimize: ${\displaystyle p\left[1-F(q^{*})\right]-c=0\Rightarrow 1-F(q^{*})={\frac {c}{p}}\Rightarrow F(q^{*})={\frac {p-c}{p}}\Rightarrow q^{*}=F^{-1}\left({\frac {p-c}{p}}\right)}$

Technically, we should also check for convexity: ${\displaystyle {\frac {\partial ^{2}}{\partial q^{2}}}\operatorname {E} [{\text{profit}}]=p[-F'(q)]}$

Since ${\displaystyle F}$ 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 ${\displaystyle (D-q)(p-c)}$ represents lost revenue not realized because of a shortage of inventory. On the other hand, if ${\displaystyle D\leq q}$, then (because the items being sold are perishable), there is an overage cost of ${\displaystyle (q-D)c}$. 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 ${\displaystyle D}$. The derivation of this is as follows:

${\displaystyle {\begin{aligned}&\operatorname {E} [{\text{opportunity cost}}+{\text{overage cost}}]\\[6pt]={}&\operatorname {E} [{\text{overage cost}}\mid D\leq q]\operatorname {P} (D\leq q)+\operatorname {E} [{\text{opportunity cost}}\mid D>q]\operatorname {P} (D>q)\\[6pt]={}&\operatorname {E} [(q-D)c\mid D\leq q]F(q)+\operatorname {E} [(D-q)(p-c)\mid D>q][1-F(q)]\\[6pt]={}&c\operatorname {E} [q-D\mid D\leq q]F(q)+(p-c)\operatorname {E} [D-q\mid D>q][1-F(q)]\\[6pt]={}&cqF(q)-c\int \limits _{x\leq q}xf(x)\,dx+(p-c)[\int \limits _{x>q}xf(x)\,dx-q(1-F(q))]\\[6pt]={}&p\int \limits _{x>q}xf(x)\,dx-pq(1-F(q))-c\int \limits _{x>q}xf(x)\,dx+cq(1-F(q))+cqF(q)-c\int \limits _{x\leq q}xf(x)\,dx\\[6pt]={}&p\int \limits _{x>q}xf(x)\,dx-pq+pqF(q)+cq-c\operatorname {E} [D]\end{aligned}}}$

The derivative of this expression, with respect to ${\displaystyle q}$, is

${\displaystyle {\frac {\partial }{\partial q}}\operatorname {E} [{\text{opportunity cost}}+{\text{overage cost}}]=p(-qf(q))-p+pqF'(q)+pF(q)+c=pF(q)+c-p}$

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:

${\displaystyle K(q)=c_{f}+c_{v}(q-x)+p\operatorname {E} \left[\max(D-q,0)\right]+h\operatorname {E} \left[\max(q-D,0)\right]}$

where the individual parameters are the following:

• ${\displaystyle c_{f}}$ – fixed cost. This cost always exists when the production of a series is started. [$/production]
• ${\displaystyle c_{v}}$ – variable cost. This cost type expresses the production cost of one product. [$/product]
• ${\displaystyle q}$ – 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.
• ${\displaystyle x}$ – initial inventory level. We assume that the supplier possesses ${\displaystyle x}$ products in the inventory at the beginning of the demand of the delivery period.
• ${\displaystyle p}$ – 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]
• ${\displaystyle D}$ – a random variable with cumulative distribution function ${\displaystyle F}$ representing uncertain customer demand. [unit]
• ${\displaystyle E[D]}$ – expected value of random variable ${\displaystyle D}$.
• ${\displaystyle h}$ – inventory and stock holding cost. [$ / product]

In ${\displaystyle K(q)}$, the first order loss function ${\displaystyle E\left[\max(D-q,0)\right]}$ captures the expected shortage quantity; its complement, ${\displaystyle E\left[\max(q-D,0)\right]}$, denotes the expected product quantity in stock at the end of the period.^[10]

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]

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