270. Stock Management

The generic structure for inventory / stock control in system dynamics: order based on expected loss + inventory gap + supply-line gap.

270.1. The classical formula

where:

• $𝑆$: current stock
• ${𝑆}^{\ast }$: target stock
• $𝑆𝐿$: current supply line (on-order, not yet received)
• $𝑆{𝐿}^{\ast }$: target supply line
• ${𝜏}_{𝑆}$: time to correct stock gap
• ${𝜏}_{𝑆𝐿}$: time to correct supply-line gap
• $𝛼\in \left[0,1\right]$: supply-line weighting (how much you account for orders already placed)
• Expected loss: replacement for expected demand

270.2. The supply-line neglect parameter

$𝛼$ is critical:

• $𝛼=1$: fully account for on-order inventory → stable, no over-ordering
• $𝛼=0$: ignore the supply line entirely → over-order while waiting → oscillation

Sterman’s beer-game experiments find $𝛼\approx 0.3$ on average — most subjects substantially neglect the supply line.

270.3. Anchoring and adjustment

Generic decision-rule structure:

• Anchor: the default level (e.g., expected demand)
• Cue: corrective signal (e.g., inventory gap)
• $𝛼$: partial-adjustment coefficient

This is the anchoring-and-adjustment heuristic (Tversky-Kahneman) operationalized as a formula. Captures bounded rationality.

270.4. Why this generates bullwhip

The chain of reasoning that produces bullwhip:

1. Retailer sees demand spike → expected demand rises
2. Inventory gap grows → orders bigger (anchor + correction)
3. Supply-line neglect → keeps ordering even though previous orders are in transit
4. Wholesaler sees retailer’s amplified orders → repeats the pattern
5. Manufacturer sees wholesaler’s even more amplified orders → repeats

Each echelon amplifies the variance. The math: see Bullwhip in SD.

270.5. Real-world example

A factory targets 100 units of inventory (${𝑆}^{\ast }=100$). Currently has 60 ($𝑆=60$). On-order 50 units ($𝑆𝐿=50$). Expected demand 20 units/period.

Settings: ${𝜏}_{𝑆}={𝜏}_{𝑆𝐿}=4$ periods, $𝛼=0.3$, $𝑆{𝐿}^{\ast }=100$ (an order-of-magnitude approximation: target supply-line equal to target stock).

270.6. See also

• Feedback Loops
• Smoothing — for expected demand
• Bullwhip in SD
• Beer Game
• Bullwhip Effect (Supply Chain)