419. Expected Profit

419.1. Expected profit and lost sales (newsvendor)

This file doesn’t relax a dimension — it computes additional outputs of the standard newsvendor model. Once you have ${𝑄}^{\ast }$, what’s the expected profit, expected leftovers, expected lost sales? The answer for normal demand uses the standard normal loss function.

419.1.1. Setup

Same as basic newsvendor: $𝐷\sim \mathrm{𝒩︀}\left(𝜇,{𝜎}^2\right)$, decision $𝑄$, prices $𝑃,𝐶,𝑆$.

Define the standard normal loss function:

where $𝜑$ is the standard normal density, $\mathrm{\Phi }$ the CDF. $𝐿\left(𝑧\right)$ measures the expected shortfall above $𝑧$ for a standard normal — equivalently, $𝐸\left[{\left(𝑍-𝑧\right)}^{+}\right]$.

Properties:

• $𝐿\left(0\right)\approx 0.399$
• $𝐿\left(𝑧\right)\to 0$ as $𝑧\to \infty$ (no shortfall expected far in the upper tail)
• $𝐿\left(𝑧\right)\to -𝑧$ as $𝑧\to -\infty$ (everything below is “missed demand” if you over-order huge)

For computation, $𝐿\left(𝑧\right)$ is tabulated or computed numerically.

419.1.2. Expected lost sales

For demand $𝐷=𝜇+𝜎𝑍$ where $𝑍\sim \mathrm{𝒩︀}\left(0,1\right)$, lost sales when $𝐷>𝑄$ amount to $𝐷-𝑄$. In standard-normal units, $𝑧=\frac{𝑄-𝜇}{𝜎}$:

This is the expected shortfall in actual units — multiply $𝐿\left(𝑧\right)$ by $𝜎$ to scale back from standard normal.

419.1.3. Expected leftovers

By symmetry, expected leftovers $=𝐸\left[{\left(𝑄-𝐷\right)}^{+}\right]$. By inventory balance:

419.1.4. Expected sales

(Total demand $𝜇$ minus the part that wasn’t satisfied.)

419.1.5. Expected profit

Per-unit revenue components:

• Sales bring in $𝑃-𝐶$ profit (margin) per unit sold.
• Leftovers: bought at $𝐶$, salvaged at $𝑆$, so loss = $𝐶-𝑆$ per leftover.

Expand:

Plug in $𝑄=𝜇+𝑧𝜎$:

The first term is the deterministic-demand profit (sell exactly $𝜇$ units at margin $𝑃-𝐶$). The second is the stochastic penalty — the cost of demand uncertainty, scaled by $𝜎$.

419.1.6. Bounded loss from uncertainty

For any $𝑧$, the penalty $𝜎\left[\left(𝑃-𝑆\right)𝐿\left(𝑧\right)+\left(𝐶-𝑆\right)𝑧\right]$ is minimized exactly when $𝑧={𝑧}^{\ast }={\mathrm{\Phi }}^{-1}\left(\text{CR}\right)$. So the optimal newsvendor minimizes the cost of uncertainty given the cost structure.