Week 5 | Session 1: News Vendor Problem — Probabilistic Demand Setting

Course: Supply Chain Digitization — Module 2: Digital Business in SC

Session Agenda

Deterministic vs. Probabilistic — What Changed?

Deterministic Setting (Sessions 4 & 5)

Demand D is known with certainty before the order is placed
Two types: (1) D is a known constant | (2) D is a known function of another variable (e.g. price)
NV dominant strategy: Always order S* = D — safe, no uncertainty

Probabilistic Setting (This Session)

D is not known with certainty — only a probability distribution of possible demand levels is known
NV must place order BEFORE demand is realised — decision is made under uncertainty
Risk of two errors: Overstock (S > D) → unsold units, procurement cost wasted | Understock (S < D) → lost sales
S* = D no longer guaranteed to be optimal — must balance overage cost vs. underage cost

Probabilistic Demand Distribution — This Example

Type: Discrete probability distribution — 5 possible demand levels Discrete: Only one scenario can realise in a given period. Scenarios are mutually exclusive. Alternative: Continuous distributions (e.g. Normal, Poisson) — same logic applies, more complex math

S#	Demand Level (D)	Probability P(D)	Interpretation
1	10,000 units	20%	Low demand scenario — 1 in 5 chance
2	20,000 units	15%	Below-average demand
3	30,000 units	10%	Moderate demand — least likely scenario
4	40,000 units	25%	Above-average demand — fairly likely
5	50,000 units	30%	High demand — most likely scenario
Total	—	100%	Discrete distribution — exactly one scenario realises per period

Note: Probabilities sum to 100% → valid distribution ✓
Most likely outcome: D = 50,000 (30% chance) → but NV cannot guarantee this will occur
Least likely: D = 30,000 (10% chance)

The Decision Problem — What Should NV Order?

NV objective: Maximise profit
NV type assumed: Risk-neutral rational player → maximises expected (average) profit, not worst-case or best-case
What NV knows in advance: The probability distribution of D — NOT the actual realised D
Decision tool: Expected value of payoffs = sum of (probability × profit) across all demand scenarios
Formula: E[Profit | S] = Σ P(Dᵢ) × π(S, Dᵢ) for all demand scenarios i

Step 1 — Sales Matrix: Units Sold = min(D, S)

Rule: Units sold = min(Demand D, Order S) — always the smaller of the two

If S > D: excess stock sits unsold → D determines sales (demand-limited)
If S < D: run out of stock before all demand is met → S determines sales (supply-limited)
If S = D: perfect match → all demand met, no leftover stock (diagonal cells below)

Order Size (S) → Demand (D) ↓	S=10K	S=20K	S=30K	S=40K	S=50K	Pattern
D=10,000	10K	10K	10K	10K	10K	D caps sales
D=20,000	10K	20K	20K	20K	20K	D caps sales
D=30,000	10K	20K	30K	30K	30K	D caps sales
D=40,000	10K	20K	30K	40K	40K	D caps sales
D=50,000	10K	20K	30K	40K	50K	S caps sales

Highlighted diagonal (S = D): maximum possible sales for that demand level
Below diagonal (D > S): supply-limited, sales = S (lost sales occur)
Above diagonal (D < S): demand-limited, sales = D (unsold stock occurs)

Step 2 — NV Profit Matrix

Formula: NV Profit = min(D, S) × ₹20 − S × ₹10

₹20 = net margin per unit sold (retail ₹30 − distribution ₹10)
S × ₹10 = total procurement cost paid to supplier (regardless of how many are sold)

Parameters	Value
Market Price (P) (Rs./unit)	30
Distribution Cost (Rs./unit)	10
Wholesale Selling Price (W) (Rs./unit)	10
Supplier’s Manufacturing & Operating Costs (Rs./unit)	5
Fixed Costs for Supplier (Rs.)	150000

Order Size (S) → Demand (D) ↓	S=10K	S=20K	S=30K	S=40K	S=50K	Logic
D=10,000	+₹1L	₹0	−₹1L	−₹2L	−₹3L	S > D → loss grows
D=20,000	+₹1L	+₹2L	+₹1L	₹0	−₹1L	Peak at S=D
D=30,000	+₹1L	+₹2L	+₹3L	+₹2L	+₹1L	Peak at S=D
D=40,000	+₹1L	+₹2L	+₹3L	+₹4L	+₹3L	Peak at S=D
D=50,000	+₹1L	+₹2L	+₹3L	+₹4L	+₹5L	Peak at S=D

Diagonal (S = D): Always the peak profit for each row (each demand level) — confirms deterministic rule
Moving right of diagonal (S > D): Profit falls — each extra unit bought but not sold costs ₹10 with no revenue
Moving left of diagonal (S < D): Profit also falls — missed sales of ₹20 each that could have been earned
S = 10K row: Always +₹1L regardless of demand — safe but leaves money on the table when D is high
S = 40K when D = 10K: −₹2L → ordering 4× demand with demand only 10K units = large wasteful procurement

Step 3 — Expected Profit Calculation (Risk-Neutral NV)

For each order size S, compute: E[Profit | S] = Σ P(Dᵢ) × NV Profit(S, Dᵢ) Excel method: SUMPRODUCT of probability vector and profit column for that order size

Order Size (S)	P=20% D=10K	P=15% D=20K	P=10% D=30K	P=25% D=40K	P=30% D=50K	E[Profit] (₹)	NV Choose?
S = 10,000	+₹1L	+₹1L	+₹1L	+₹1L	+₹1L	₹1,00,000	No
S = 20,000	₹0	+₹2L	+₹2L	+₹2L	+₹2L	₹1,60,000	No
S = 30,000	−₹1L	+₹1L	+₹3L	+₹3L	+₹3L	₹1,90,000	No
S = 40,000 ★	−₹2L	₹0	+₹2L	+₹4L	+₹4L	₹2,00,000 ★	YES ✓
S = 50,000	−₹3L	−₹1L	+₹1L	+₹3L	+₹5L	₹1,50,000	No

Step-by-Step Examples

E[Profit | S = 10,000]: Same profit (+₹1L) across all demand levels → E[Profit] = 20%×1L + 15%×1L + 10%×1L + 25%×1L + 30%×1L = ₹1,00,000. Safe choice but misses upside when demand is high.
E[Profit | S = 20,000]: ₹0 when D=10K (no profit no loss) | +₹2L for D ≥ 20K. E[Profit] = 20%×0 + (15%+10%+25%+30%)×2L = 0.8 × 2,00,000 = ₹1,60,000
E[Profit | S = 40,000] — OPTIMAL ★: Profits: D=10K → −₹2L | D=20K → ₹0 | D=30K → +₹2L | D=40K → +₹4L | D=50K → +₹4L. E[Profit] = 20%×(−2L) + 15%×0 + 10%×2L + 25%×4L + 30%×4L = −40,000 + 0 + 20,000 + 1,00,000 + 1,20,000 = ₹2,00,000 ← highest ✓

Key Result — Probabilistic Setting

NV optimal order: S* = 40,000 units → E[Profit] = ₹2,00,000
Important: S* = 40,000 ≠ any specific demand level. It is NOT simply “match D”
The NV is willing to risk ordering 40K even though it sometimes leads to loss (when D = 10K) — because the expected value across all scenarios is highest at S = 40K
Ordering S = 50K reduces E[Profit] back to ₹1,50,000 — overstocking penalty outweighs high-demand upside

Why S > expected demand?* High-demand scenarios (D=40K, 30%; D=50K, 25%) are probable enough that the upside of extra stock outweighs the downside of overstocking in the low-demand case. Tradeoff at core of NVP: Overage cost (cost of ordering too much) vs. Underage cost (cost of ordering too little)

Deterministic vs. Probabilistic — Side-by-Side

Aspect	Deterministic	Probabilistic
Demand knowledge	Known exactly before order	Known only as a probability distribution
NV decision rule	S* = D (always match demand)	S* = arg max E[Profit \| S]
Risk	None — D is certain	Overstock loss OR lost sales — cannot be avoided
S vs. demand*	S* = D (exact match)	S* may differ from any specific D (here: S*=40K, no D=40K certain)
NV expected profit	Exact profit known in advance	₹2,00,000 (expected, not guaranteed)
SC conflict?	Yes — when D < RS breakeven (30K)	To be analysed in next session

What Comes Next — Preview

Next session: Supplier payoff analysis under probabilistic setting — does S*=40K make RS profitable?
Then: contracts as a coordination tool — simple contracts + popular contracts (e.g. buyback, revenue sharing)
Finally: how platform economy facilitates coordination — the bridge from NVP to real-world SC digitization

Session Summary

Probabilistic NVP: D is uncertain — NV knows only the probability distribution, not the actual D
Demand distribution: 5 discrete scenarios: 10K(20%), 20K(15%), 30K(10%), 40K(25%), 50K(30%)
Sales rule: Units sold = min(D, S) — always
NV profit per cell: min(D,S) × ₹20 − S × ₹10
Decision framework: Risk-neutral NV maximises E[Profit | S] = Σ P(Dᵢ) × profit(S, Dᵢ)
Result: S* = 40,000 units → E[Profit] = ₹2,00,000 (highest across all options)
Key insight: S* ≠ D in probabilistic setting. Determined by balancing overage vs. underage costs across all scenarios.