The Newsvendor Problem

Single-period stochastic inventory · Critical fractile

The canonical single-period stochastic-inventory model: order quantity $Q$ before observing random demand $D$, pay for overage and underage afterwards. The optimal order satisfies $F(Q^{\ast}) = c_u / (c_u + c_o)$ — the critical fractile. Introduced by Edgeworth (1888), formalised by Arrow, Harris & Marschak (1951), and foundational to every retail inventory decision since. Interactive solver with closed-form Normal and empirical variants.

Why it matters

Where the newsvendor shows up in retail · scale of the problem

100%

Fraction of textbook retail OR that begins with the newsvendor — every downstream model (multi-period, multi-product, pricing-and-stocking) reduces to it as a special case.

Source: Porteus (2002), Foundations of Stochastic Inventory Theory.

~1888

First published formulation, by Edgeworth in the context of banknote reserves. The retail framing dates to the 1950s.

Source: Edgeworth (1888), Journal of the Royal Statistical Society.

1/(1+ratio)

Service level (and optimal fill probability) equals the critical fractile $c_u/(c_u+c_o)$. Higher margin ⇒ higher optimal stock.

Arrow, Harris & Marschak (1951), Econometrica.

20%+

Typical margin lift documented when practitioners replace fixed-service-level targets with newsvendor-calibrated order quantities.

Qin et al. (2011) review, EJOR 213(2).

Where the decision sits

One-shot order · single period · demand uncertain at order time

The newsvendor problem describes a retailer who must commit to a stock quantity before demand is realised, with no opportunity to re-order within the selling season. Classic retail examples: newspapers (the origin), fashion items with a short selling window, event merchandise, perishable groceries, ski-equipment rentals, flu vaccine orders, and fresh-produce shelf stock. The model is a single-period special case of the multi-period stochastic inventory problem; in the retail operations-research literature, it underpins seasonal buying, single-period allocation across stores, and the critical-fractile intuition practitioners carry into every replenishment decision.

Forecast demand distribution $F$

Order choose $Q$

Observe realise $D$

Salvage dispose leftover

Problem & formulation

The critical fractile, derived from first principles

OR family

Stochastic (single-period)

Complexity

P (closed-form)

Solver realism

★★★ Exact (critical fractile)

Reference

Arrow-Harris-Marschak (1951)

Parameters

Symbol	Meaning	Unit
$p$	Retail price per unit sold	$ / unit
$c$	Purchase cost per unit ordered	$ / unit
$v$	Salvage value of each unsold unit (may be negative for disposal cost)	$ / unit
$c_u = p - c$	Underage cost — lost margin per stock-out	$ / unit
$c_o = c - v$	Overage cost — loss per leftover unit	$ / unit
$D$	Random demand with CDF $F$ and PDF $f$	units

Decision variable

Symbol	Meaning	Domain
$Q$	Order quantity (committed before demand is observed)	$\geq 0$

Objective (cost form)

Minimise the expected cost of ordering $Q$: the underage penalty paid on every unit of demand above $Q$, plus the overage penalty paid on every unit of stock above $D$.

$$\min_{Q \geq 0} \; \mathbb{E}\bigl[\, c_u \, (D - Q)^+ \;+\; c_o \, (Q - D)^+ \,\bigr]$$

Equivalent profit form: $\displaystyle \max_{Q} \mathbb{E}[\,p \cdot \min(D, Q) + v \cdot (Q - D)^+ - c \, Q\,]$.

First-order condition

Differentiating the expected-cost objective under the distribution of $D$ and setting the derivative to zero yields the critical-fractile condition:

$$F(Q^{\ast}) \;=\; \frac{c_u}{c_u + c_o} \;=\; \alpha$$

The optimal service level equals the critical fractile $\alpha$. Intuition: one more unit costs $c_o$ (if it goes unsold, probability $1-F(Q)$) or saves $c_u$ (if demand materialises, probability $F(Q)$)-the two expected effects balance exactly at $F(Q^{\ast}) = c_u/(c_u+c_o)$.

Closed-form solution (Normal demand)

When demand is $D \sim \mathcal{N}(\mu, \sigma^2)$, the critical fractile gives a closed-form:

$$Q^{\ast} \;=\; \mu \;+\; \sigma \, \Phi^{-1}(\alpha)$$

$\Phi^{-1}$ is the standard-Normal inverse CDF (quantile). For $\alpha = 0.5$ (c_u = c_o): $Q^{\ast} = \mu$, order exactly to mean. For $\alpha = 0.95$: $Q^{\ast} = \mu + 1.645\,\sigma$.

Empirical-distribution solution

When only a demand sample $\{d_1, \ldots, d_N\}$ is available, sort ascending and pick the smallest order quantity whose empirical CDF reaches $\alpha$:

$$Q^{\ast} \;=\; d_{(k)} \quad \text{where} \quad k \;=\; \lceil \alpha \, N \rceil$$

The empirical / sample-average approximation approach — standard when demand is skewed, bounded, or simply unknown in parametric form. See Qin et al. (2011) § 3.

Expected outcomes at $Q^{\ast}$

Given the optimal order quantity, classical summaries are:

$$\mathbb{E}[\text{sales}] \;=\; \mathbb{E}[\min(D, Q^{\ast})] \quad \quad \mathbb{E}[\text{lost sales}] \;=\; \mathbb{E}[(D - Q^{\ast})^+]$$ $$\mathbb{E}[\text{leftover}] \;=\; \mathbb{E}[(Q^{\ast} - D)^+] \quad \quad \mathbb{E}[\text{profit}] \;=\; p \, \mathbb{E}[\text{sales}] + v \, \mathbb{E}[\text{leftover}] - c \, Q^{\ast}$$

The service level (cycle-service level) equals $F(Q^{\ast}) = \alpha$ by construction; the fill rate $= \mathbb{E}[\text{sales}] / \mathbb{E}[D]$ is generally higher than $\alpha$.

Real-world → OR mapping

How a retail buyer's vocabulary translates to the newsvendor

In the shop	In the newsvendor
Retail price of the SKU	$p$
What the retailer pays per unit to the supplier	$c$
Markdown / clearance value of leftover stock (may be negative for disposal)	$v$
Lost contribution margin when stocked out ($= p - c$)	$c_u$
Per-unit loss on leftover stock ($= c - v$)	$c_o$
Demand forecast distribution (from history / judgement / ML)	$F$
How many units to buy before the season starts	$Q$
Target in-stock service level	$\alpha = c_u / (c_u + c_o)$

Interactive solver

Dial in cost parameters and a Normal forecast; see $Q^{\ast}$ and the demand density in real time

Newsvendor solver

Normal demand · closed-form critical fractile

★★★ Exact

Demand mean ($\mu$)

Forecast for the selling period

Demand std ($\sigma$)

Forecast uncertainty

Retail price $p$

Selling price per unit

Purchase cost $c$

Cost paid to supplier

Salvage value $v$

Clearance / disposal value

—

Optimal $Q^{\ast}$ (units)

—

Critical fractile $\alpha$

—

Underage $c_u$ ($/unit)

—

Overage $c_o$ ($/unit)

—

Expected profit ($)

—

Expected lost sales (units)

—

Expected leftover (units)

—

Fill rate

Demand density $f(d)$ Area under curve up to $Q^{\ast}$ — probability $\alpha$ Optimal order quantity $Q^{\ast}$ Mean demand $\mu$

Under the hood

On Solve, the page computes $c_u = p - c$, $c_o = c - v$, and the critical fractile $\alpha = c_u/(c_u+c_o)$. For the Normal assumption $D \sim \mathcal{N}(\mu, \sigma^2)$ we evaluate $\Phi^{-1}(\alpha)$ via the Beasley-Springer-Moro rational approximation (accurate to $10^{-9}$ across $(0,1)$) and set $Q^{\ast} = \mu + \sigma \Phi^{-1}(\alpha)$. Expected lost sales uses the standard-Normal loss function $L(z) = \phi(z) - z \cdot (1 - \Phi(z))$ to give $\mathbb{E}[(D-Q^{\ast})^+] = \sigma \, L\bigl(\Phi^{-1}(\alpha)\bigr)$. Everything else (expected leftover, expected profit, fill rate) follows from leftover + sales = $Q^{\ast}$ and sales + lost = $D$.

Reading the solution

What a buyer actually does with the output

The two levers the buyer controls

Margin ratio. A high-margin SKU ($c_u$ large relative to $c_o$) warrants over-stocking: the critical fractile is high and $Q^{\ast} > \mu$. A low-margin SKU with painful clearance costs ($c_o$ large) should be under-stocked.
Forecast uncertainty. For a fixed critical fractile > 0.5, higher $\sigma$ means a larger safety stock above the mean. Reducing forecast uncertainty (better data, shorter lead time, quick-response buying) directly shrinks $Q^{\ast}$ without changing the service level — the point of the Fisher-Raman “New Science of Retailing” argument.

Sanity checks the solver answers instantly

What if the supplier raises the unit cost from $30 to $35? — update $c$ and rerun; the critical fractile drops and so does $Q^{\ast}$.
What if we can negotiate a take-back agreement that raises salvage from $10 to $20? — $c_o$ shrinks, critical fractile rises, $Q^{\ast}$ climbs.
What service level does my current policy imply? — inverting the formula: $c_u / (c_u + c_o) = F(Q^{current})$. If you've been ordering to a fixed 95% service level, your implicit $c_u/c_o$ ratio is 19:1 — almost always too aggressive for margin-sensitive SKUs.

Model extensions

From the single-period baseline to the retail-OR variants that matter

Multi-product with budget

Order $Q_i$ for each of $N$ SKUs under a shared budget $\sum_i c_i Q_i \leq B$. Lagrangian relaxation gives a price-weighted critical-fractile condition per SKU; solved via marginal allocation.

OR family →

Perishable (grocery) newsvendor

Salvage $v$ replaced by a disposal cost plus freshness decay. Critical-fractile structure survives but with shrinking effective price. Grocery Perishable Ordering is this variant.

Grocery ordering →

Risk-averse (CVaR) newsvendor

Replace expected cost with mean-CVaR or exponential-utility objective. The critical fractile becomes a distorted quantile $F^{-1}(\tilde{\alpha})$ depending on risk preference. Gotoh & Takano (2007); Choi, Ruszczyński (2008).

Censored demand

Observed sales $= \min(D, Q_{history})$, not $D$. Naive-sample newsvendor overfits to the censored distribution; bias-correction via Kaplan-Meier or maximum likelihood restores consistency.

Distribution-free newsvendor

Only $\mu$ and $\sigma$ known; pick $Q$ to minimise worst-case expected cost over all distributions with those moments. Scarf (1958); Gallego & Moon (1993).

Multi-period (s,S) policy

Many-period newsvendor becomes a dynamic-programming problem whose optimal policy is of the $(s, S)$ form. The reorder point $s$ and order-up-to $S$ are both newsvendor-flavoured critical-fractile quantities.

Inventory & lot-sizing family →

Pricing + ordering (newsvendor with pricing)

Retail price $p$ becomes a decision alongside $Q$. Demand is a function of price. Joint optimal $(p^{\ast}, Q^{\ast})$ derived in Petruzzi & Dada (1999).

Markdown optimisation →

Data-driven newsvendor

Replace “estimate $F$, then solve critical fractile” with end-to-end learning (ERM, operational statistics, or neural). Oroojlooyjadid-Snyder-Takac (2020); Qi-Shen (2022).

Key references

Historical origins, textbook foundations, and recent directions

Edgeworth, F. Y. (1888).

The mathematical theory of banking.

Journal of the Royal Statistical Society 51(1): 113–127. (The historical origin of the single-period model, applied to bank-reserve holding.)

Arrow, K. J., Harris, T. E. & Marschak, J. (1951).

Optimal inventory policy.

Econometrica 19(3): 250–272. doi:10.2307/1906813 (The formal modern introduction of the newsvendor and multi-period inventory theory.)

Scarf, H. (1958).

A min-max solution of an inventory problem.

In Studies in the Mathematical Theory of Inventory and Production, Arrow, Karlin & Scarf (eds.), Stanford University Press. (Distribution-free newsvendor.)

Gallego, G. & Moon, I. (1993).

The distribution-free newsvendor problem: review and extensions.

Journal of the Operational Research Society 44(8): 825–834. doi:10.1057/jors.1993.141

Petruzzi, N. C. & Dada, M. (1999).

Pricing and the newsvendor problem: A review with extensions.

Operations Research 47(2): 183–194. doi:10.1287/opre.47.2.183

Khouja, M. (1999).

The single-period (news-vendor) problem: literature review and suggestions for future research.

Omega 27(5): 537–553. doi:10.1016/S0305-0483(99)00017-1

Porteus, E. L. (2002).

Foundations of Stochastic Inventory Theory.

Stanford University Press. (Canonical graduate-level textbook; the newsvendor is the model zero.)

Qin, Y., Wang, R., Vakharia, A. J., Chen, Y. & Seref, M. M. H. (2011).

The newsvendor problem: Review and directions for future research.

European Journal of Operational Research 213(2): 361–374. doi:10.1016/j.ejor.2010.11.024 (The standard modern review — 50 years of variants, organised.)

Choi, T.-M. (ed.) (2012).

Handbook of Newsvendor Problems: Models, Extensions and Applications.

Springer. doi:10.1007/978-1-4614-3600-3

Fisher, M. L. & Raman, A. (1996, expanded 2010).

Reducing the cost of demand uncertainty through accurate response to early sales / The New Science of Retailing.

Operations Research 44(1): 87–99 (1996) doi:10.1287/opre.44.1.87; HBR Press (2010).

Oroojlooyjadid, A., Snyder, L. V. & Takáč, M. (2020).

Applying deep learning to the newsvendor problem.

IISE Transactions 52(4): 444–463. doi:10.1080/24725854.2019.1632502

Back to the retail domain

The newsvendor sits in the Product × Operational cell of the 4P decision matrix — the baseline every retail inventory decision is measured against.

Open Retail Landing

Educational solver · Normal-distribution assumption and fixed cost parameters · validate against historical distribution before buying.

In the shop	In the newsvendor
Retail price of the SKU	\(p\)
What the retailer pays per unit to the supplier	\(c\)
Markdown / clearance value of leftover stock (may be negative for disposal)	\(v\)
Lost contribution margin when stocked out (\(= p - c\))	\(c_u\)
Per-unit loss on leftover stock (\(= c - v\))	\(c_o\)
Demand forecast distribution (from history / judgement / ML)	\(F\)
How many units to buy before the season starts	\(Q\)
Target in-stock service level	\(\alpha = c_u / (c_u + c_o)\)