Bullwhip SD

271. Bullwhip SD

The quantitative analysis of supply-chain order amplification, from the system-dynamics perspective. Closed-form result by Chen, Drezner, Ryan, Simchi-Levi (2000) for the order-up-to (OUT) policy with AR(1) demand.

271.1. Chen-Drezner-Ryan-Simchi-Levi formula

For demand $𝐷_{𝑡}$ following an AR(1) process and an OUT policy with smoothing parameter $𝑝$ and lead time $𝐿$ :

\frac{Var (orders)}{Var (demand)} = 1 + \frac{2 𝐿}{𝑝} + \frac{2 𝐿^{2}}{𝑝^{2}}

The order variance is always at least the demand variance, and grows quadratically with $\frac{𝐿}{𝑝}$ ratio. For two-echelon chains, this multiplies stage by stage.

271.2. Key insights

Lead time matters most: doubling $𝐿$ roughly quadruples the bullwhip
Smoothing matters too: longer smoothing ( $𝑝$ ) reduces bullwhip but adds delay
No exact balance: even optimal $𝑝$ doesn’t achieve $Var (𝑂) = Var (𝐷)$ — always amplified

271.3. Sources of bullwhip (Lee-Padmanabhan-Whang 1997)

Four causes, each contributing variance amplification:

Demand signal processing: each echelon uses smoothed orders (not actual demand) — like the Chen-DRSL formula
Order batching: large infrequent orders smooth retailer’s demand into spikes upstream
Price fluctuations: forward-buying during promotions creates spikes
Rationing & shortage gaming: customers over-order during shortages, creating ghost demand

Each cause is amplified by long lead times.

271.4. SD beer-game replication

The Sterman beer-game model implements this dynamically:

Each echelon’s decision-rule (anchoring-adjustment with supply-line neglect) propagates variance
Initial customer-demand step (small) gets amplified at each echelon
Final factory order is 5-10x the customer-demand variance — matching experimental data

This is the quantitative validation of the SD bullwhip explanation.

271.5. Mitigations

What reduces bullwhip:

Short lead times: $𝐿$ down → quadratic reduction
Information sharing: upstream sees real customer demand, not amplified orders (CPFR, VMI)
Order smoothing: don’t fully correct gaps instantly
Stable pricing: avoid promotion-induced spikes
Capacity allocation rules: use historical demand, not current orders (avoid rationing game)

271.6. See also

Stock Management — the decision rule
Beer Game — simulation
Bullwhip Effect (Supply Chain) — broader view
Delays — what enables amplification