Discrete

418. Discrete

418.1. Newsvendor with discrete demand

Relax one dimension from basic newsvendor: demand distribution is no longer continuous. Demand $𝐷$ 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.

418.1.1. Setup

Same costs as basic newsvendor:

$𝐶_{𝑢} = 𝑃 - 𝐶$ (underage)
$𝐶_{𝑜} = 𝐶 - 𝑆$ (overage)
$CR = 𝐶_{𝑢} / (𝐶_{𝑢} + 𝐶_{𝑜})$

Demand $𝐷 \in {0, 1, 2, \dots}$ with PMF $𝑝_{𝑑}$ and CDF $𝐹 (𝑑) = 𝑃 (𝐷 \leq 𝑑) = \sum_{𝑘 = 0}^{𝑑} 𝑝_{𝑘}$ .

418.1.2. Decision rule

For continuous $𝐹$ , $𝑄^{*} = 𝐹^{- 1} (CR)$ . For discrete $𝐹$ , the CDF jumps — there may be no $𝑄$ with $𝐹 (𝑄) = CR$ exactly. Take the smallest $𝑄$ that gets you to (or above) CR:

𝑄^{*} = min {𝑄 \in {0, 1, 2, \dots} : 𝐹 (𝑄) \geq CR}

Equivalent statements:

$𝐹 (𝑄^{*}) \geq CR$ (don’t undershoot)
$𝐹 (𝑄^{*} - 1) < CR$ (smallest such $𝑄$ )

Why this is correct: the marginal-analysis derivation from the basic newsvendor still holds. The expected marginal profit of one more unit at level $𝑄$ is $(1 - 𝐹 (𝑄)) 𝐶_{𝑢} - 𝐹 (𝑄) 𝐶_{𝑜}$ . Order while this is $\geq 0$ , i.e., while $𝐹 (𝑄) \leq CR$ . Stop at the first $𝑄$ where $𝐹 (𝑄) \geq CR$ .

Example: Empirical demand from sales records

Given: 6 weeks of sales for a perishable item.

Sale price: $𝑃$ = $10
Purchase cost: $𝐶$ = $4
Salvage value: $𝑆$ = $1
Empirical demand counts:

Demand $𝑑$	10	12	14	16	18	20
Frequency	1	1	2	1	0	1

Total weeks: 6.

Step 1 — costs and CR

𝐶_{𝑢} = 10 - 4 = 6, 𝐶_{𝑜} = 4 - 1 = 3

CR = \frac{6}{6 + 3} = \frac{2}{3} \approx 0.667

Step 2 — empirical CDF

Convert frequencies to probabilities, then accumulate:

$𝑑$	10	12	14	16	18	20
$𝑝_{𝑑}$	$1 / 6$	$1 / 6$	$2 / 6$	$1 / 6$	$0$	$1 / 6$
$𝐹 (𝑑)$	$1 / 6 \approx 0.167$	$2 / 6 \approx 0.333$	$4 / 6 \approx 0.667$	$5 / 6 \approx 0.833$	$5 / 6$	$6 / 6 = 1$

Step 3 — find smallest $𝑄$ with $𝐹 (𝑄) \geq 0.667$

$𝐹 (10) = 0.167 < 0.667$
$𝐹 (12) = 0.333 < 0.667$
$𝐹 (14) = 0.667 \geq 0.667$ ✓ — first to satisfy.

𝑄^{*} = 14 units

Step 4 — sanity check via marginal analysis

At $𝑄 = 14$ , expected marginal profit of one more unit:

Δ (14) = (1 - 0.667) \cdot 6 - 0.667 \cdot 3 = 2 - 2 = 0 ✓

Adding a 15th unit would yield $Δ (15) = (1 - 0.833) \cdot 6 - 0.833 \cdot 3 = 1 - 2.5 < 0$ — not worth it.

Compare to continuous-normal baseline: empirical mean $\approx 14$ , std $\approx 3.4$ . If we had assumed normal demand:

𝑄_{normal}^{*} \approx 14 + 0.44 \cdot 3.4 \approx 15.5 \to 16 (rounded)

The discrete answer ( $𝑄^{*} = 14$ ) 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.