414. Discrete

414.1. Newsvendor with discrete demand

Relax one dimension from basic newsvendor: demand distribution is no longer continuous. Demand takes integer values (Poisson, empirical histograms, etc.). The critical-ratio rule still applies, but the quantile is found by a discrete lookup instead of inverting a continuous CDF.

414.1.1. Setup

Same costs as basic newsvendor:

Demand with PMF and CDF .

414.1.2. Decision rule

For continuous , . For discrete , the CDF jumps — there may be no with exactly. Take the smallest that gets you to (or above) CR:

Equivalent statements:

Why this is correct: the marginal-analysis derivation from the basic newsvendor still holds. The expected marginal profit of one more unit at level is . Order while this is , i.e., while . Stop at the first where .

Example: Empirical demand from sales records

Given: 6 weeks of sales for a perishable item.

  • Sale price: = $10
  • Purchase cost: = $4
  • Salvage value: = $1
  • Empirical demand counts:
Demand 10 12 14 16 18 20
Frequency 1 1 2 1 0 1

Total weeks: 6.

Step 1 — costs and CR

Step 2 — empirical CDF

Convert frequencies to probabilities, then accumulate:

10 12 14 16 18 20

Step 3 — find smallest with

  • ✓ — first to satisfy.

Step 4 — sanity check via marginal analysis

At , expected marginal profit of one more unit:

Adding a 15th unit would yield — not worth it.

Compare to continuous-normal baseline: empirical mean , std . If we had assumed normal demand:

The discrete answer () is smaller than the continuous approximation because the empirical CDF jumps past the threshold exactly at 14. With small samples, the discrete formula respects the actual data distribution rather than a smoothed assumption.