269. 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.
269.1. Chen-Drezner-Ryan-Simchi-Levi formula
For demand following an AR(1) process and an OUT policy with smoothing parameter and lead time :
The order variance is always at least the demand variance, and grows quadratically with ratio. For two-echelon chains, this multiplies stage by stage.
269.2. Key insights
- Lead time matters most: doubling roughly quadruples the bullwhip
- Smoothing matters too: longer smoothing () reduces bullwhip but adds delay
- No exact balance: even optimal doesn’t achieve — always amplified
269.3. 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.
269.4. 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.
269.5. Mitigations
What reduces bullwhip:
- Short lead times: 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)
269.6. See also
- Stock Management — the decision rule
- Beer Game — simulation
- Bullwhip Effect (Supply Chain) — broader view
- Delays — what enables amplification