10. Newsvendor Model

The newsvendor (or newsboy) model answers a single-period question: how much should I produce or order before demand is known, when you cannot replenish during the selling period and leftover units are costly?

Classic stories include daily newspapers, fashion seasons, event catering, and bakery items baked overnight. This lesson builds the model from a discrete muffin shop example, then generalizes overage and underage costs, expected profit, and the critical ratio for continuous and discrete demand.

Original teaching rewrite for this course; probability tables and numeric drills are constructed for practice, not copied from a commercial textbook.

One-period bakery: chocolate muffins

Every evening you choose how many chocolate muffins to bake for tomorrow. Baking takes too long to react to same-day demand. From history you estimate discrete daily demand $D$ (integer units only):

$d$	0	2	4	6	8	10
$p(d)$	0.40	0.20	0.20	0.10	0.05	0.05
$P(D\le x)$	0.40	0.60	0.80	0.90	0.95	1.00

Notation:

• $p(d)$ — probability mass function (PMF): $\Pr(D=d)$.
• $P(D\le x)$ — cumulative distribution function (CDF) for discrete $D$ (same role as $F_D(x)$ for continuous demand).

Economics for one day:

• Unit cost $c=2$ (bake)
• Selling price $p=6$
• Unsold muffins are cleared at salvage value $s_v=1$ (discounted end-of-day price)

Salvage value is whatever you recover (or pay) when clearing leftover inventory—discounted sale, supplier buyback, or disposal cost.

Profit for one demand outcome

Let $Q$ be the bake quantity and $d$ the realized demand. Sales are $\min(Q,d)$; leftovers are $\max(0,Q-d)$.

$$\mathcal{P}(Q,d)=p\,\min(Q,d)+s_v\,\max(0,Q-d)-c\,Q$$

Revenue from sales plus salvage on excess, minus full production cost for $Q$ units.

Expected profit by $Q$

$$\mathcal{P}(Q)=\sum_d \mathcal{P}(Q,d)\,p(d)$$

For candidate quantities $Q\in\{2,4,6,8,10\}$, expected profits are about 4, 6, 6, 5, 3.5 currency units. $Q=4$ and $Q=6$ tie for the maximum (6); $Q=4$ is the smallest maximizer among that grid. A full profit table by $(Q,d)$ is the same idea as a spreadsheet SUMPRODUCT over rows of demand.

When is this a newsvendor problem?

Four structural conditions:

1. The order (or production run) is placed before the season or selling day.
2. No replenishment during the season.
3. Overage cost if you prepare too much.
4. Underage cost if you prepare too little (lost sales, penalties, or goodwill).

Examples: seasonal apparel, calendars, champagne for an event, and—at a higher level—safety stock over one replenishment cycle when the “season” is the time until the next order.

Overage and underage unit costs

Split mismatches into two unit costs (positive numbers when they represent a cost):

	Meaning	Typical muffin/cookie form
Overage $c_o$	Cost of one unit left at period end	$c_o=c-s_v$ when unsold units are salvaged (not held to next period)
Underage $c_u$	Cost of one lost sale (profit + penalties)	$c_u=p-c+g$; goodwill/penalty $g\ge 0$ optional

If leftover units can be held to the next period at holding cost $h$, use $c_o=h$ instead of full write-off.

When the model minimizes cost, include lost profit in $c_u$. When it maximizes profit as in $\mathcal{P}(Q,d)$ above, do not double-count opportunity cost inside $c_u$.

For the muffins: $c_o=2-1=1$, $c_u=6-2=4$ (no goodwill).

Expected sales, excess, and shortage

For each $(Q,d)$:

Outcome	$(Q,d)$	Discrete expectation in $Q$
Sales	$\min(Q,d)$	$\sum_{d\le Q} d\,p(d)$
Excess units	$\max(0,Q-d)$	$\sum_{d<Q}(Q-d)\,p(d)$
Units short	$\max(0,d-Q)$	$\sum_{d>Q}(d-Q)\,p(d)$

General profit decomposition (penalty/goodwill in $c_u$ only when using the cost form):

$$\mathcal{P}(Q,d)=p\,\min(Q,d)-c_o\,\max(0,Q-d)-c_u\,\max(0,d-Q)-c\,Q$$

Summing over $p(d)$ gives $\mathcal{P}(Q)$ in terms of $Q$ alone.

Critical ratio and optimal order quantity

Define the critical ratio (target cycle service level at optimum):

$$\alpha^*=\frac{c_u}{c_u+c_o}$$

Continuous demand

For continuous demand with CDF $F_D$, marginal balance on the $Q$-th unit gives:

$$c_o F_D(Q)=c_u\bigl(1-F_D(Q)\bigr) \quad\Rightarrow\quad F_D(Q^*)=\alpha^* \quad\Rightarrow\quad Q^*=F_D^{-1}(\alpha^*)$$

Cookie sketch (Normal): $p=3$, $c=1$, no salvage → $c_u=2$, $c_o=1$, $\alpha^*=2/3\approx 66.7\%$. If demand is $\mathcal{N}(\mu=15,\sigma=5)$, then $Q^*\approx \mu+z_{0.67}\sigma\approx 17.2$—bake 17 if you round down to integer pans.

Discrete demand

Increase $Q$ to $Q+1$ while expected underage savings beat overage:

$$c_o P(D\le Q)\le c_u P(D>Q) \;\Rightarrow\; P(D\le Q)\le \alpha^*$$

Stop at the smallest integer $Q$ with

$$P(D\le Q)\ge \alpha^*$$

For muffins, $\alpha^*=4/5=0.8$. On the lesson PMF, $P(D\le 4)=0.80$, so the critical-ratio rule selects $Q^*=4$—the same smallest maximizer as the expected-profit grid ($Q=4$ and $Q=6$ both earn 6 in expectation).

Cookie sketch (discrete PMF, substitute product): if each lost cookie still yields 1 in alternate margin, effective $c_u=p-c-1=1$, still $c_o=1$, $\alpha^*=50\%$. With a refined PMF on even counts, $Q=14$ is the smallest quantity with $P(D\le 14)\ge 0.5$ (75% CDF), matching a full profit table.

Cost-only form

Expected cost (with $c_u$ including lost profit per unit short):

$$\mathcal{C}(Q)=c_o\sum_{d<Q}(Q-d)\,p(d)+c_u\sum_{d>Q}(d-Q)\,p(d)$$

Minimizing $\mathcal{C}(Q)$ is equivalent to maximizing profit when definitions align. At the optimum, total expected overage and underage costs balance—a useful Excel sanity check.

Practice

Concept check

Quiz

Newsvendor structure and costs

1 / 5

Which situation is NOT consistent with a standard newsvendor setup?

Pick an answer to see if you are right.

Rank the discrete newsvendor workflow

Quiz

Plan a one-period quantity

1 / 1

You have a discrete demand PMF and unit economics for a single selling period.

OrderDrag the handle to rearrange — top should be first.

1. ⋮⋮1Build or verify the CDF P(D ≤ Q) from the PMF.
2. ⋮⋮2Compute the critical ratio α* = c_u / (c_u + c_o).
3. ⋮⋮3Estimate unit overage c_o and underage c_u (salvage, goodwill, or holding as appropriate).
4. ⋮⋮4Choose the smallest Q with P(D ≤ Q) ≥ α*, then cross-check expected profit on a candidate grid.
5. ⋮⋮5Confirm the decision is one-period with no in-season replenishment.

Drag items into the correct order (or use ↑↓), then check.

Code: expected profit on a quantity grid

Implement expected newsvendor profit for discrete demand. Use the profit form with salvage (same as the muffin spreadsheet). On a profit tie, return the smallest quantity among the candidates—even if the list is not sorted.

Code: discrete critical ratio

Return the smallest integer order quantity $Q$ such that the discrete CDF satisfies $P(D\le Q)\ge c_u/(c_u+c_o)$. Treat demands and pmf as parallel lists (PMF over those support points). Search $Q$ from 0 upward (you may assume some $Q$ up to max(demands)+1 suffices).

Takeaways

• Newsvendor fits one-period, no-replenishment decisions with both overage and underage mismatch costs.
• Discrete demand uses a PMF and CDF; continuous demand uses a density and $F_D^{-1}$.
• Critical ratio $\alpha^*=c_u/(c_u+c_o)$ sets optimal service: $Q^*=F_D^{-1}(\alpha^*)$ (continuous) or smallest $Q$ with $P(D\le Q)\ge\alpha^*$ (discrete).
• Expected profit $\mathcal{P}(Q)=\sum_d \mathcal{P}(Q,d)p(d)$ is a universal check when the PMF is custom or messy.
• Mismatch costs can include salvage revenue, holding, goodwill, or penalties—define $c_o$ and $c_u$ consistently with whether you maximize profit or minimize cost.

Previous: Distribution MEIO and Serial Simulation.

Next: Discrete Probabilistic Demand — histograms, Gaussian KDE, and a simulation-ready PMF.