415. Expected Profit

415.1. 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 , what’s the expected profit, expected leftovers, expected lost sales? The answer for normal demand uses the standard normal loss function.

415.1.1. Setup

Same as basic newsvendor: , decision , prices .

Define the standard normal loss function:

where is the standard normal density, the CDF. measures the expected shortfall above for a standard normal — equivalently, .

Properties:

• as (no shortfall expected far in the upper tail)
• as (everything below is “missed demand” if you over-order huge)

For computation, is tabulated or computed numerically.

415.1.2. Expected lost sales

For demand where , lost sales when amount to . In standard-normal units, :

This is the expected shortfall in actual units — multiply by to scale back from standard normal.

415.1.3. Expected leftovers

By symmetry, expected leftovers . By inventory balance:

415.1.4. Expected sales

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

415.1.5. Expected profit

Per-unit revenue components:

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

Expand:

Plug in :

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

415.1.6. Bounded loss from uncertainty

For any , the penalty is minimized exactly when . So the optimal newsvendor minimizes the cost of uncertainty given the cost structure.