---

name: "algo-sc-eoq" description: "Calculate Economic Order Quantity to minimize total inventory cost (ordering + holding). Use this skill when the user needs to determine optimal order size, balance ordering frequency against storage costs, or set reorder points — even if they say 'how much to order', 'optimal batch size', or 'inventory cost minimization'." metadata: category: "WP-41 供應鏈演算法" tags: ["supply-chain", "eoq", "inventory", "ordering"]

Economic Order Quantity (EOQ)

Overview

EOQ determines the order quantity that minimizes total inventory cost = ordering cost + holding cost. Formula: EOQ = √(2DS/H) where D=annual demand, S=ordering cost per order, H=holding cost per unit per year. Assumes constant demand and instantaneous replenishment.

When to Use

Trigger conditions:

• Setting standard order quantities for inventory replenishment
• Balancing ordering frequency against warehousing costs
• Baseline calculation before applying safety stock adjustments

When NOT to use:

• When demand is highly uncertain (use newsvendor model)
• When products are perishable with short shelf life
• When quantity discounts change the cost structure significantly

Algorithm

IRON LAW: EOQ Assumes CONSTANT, KNOWN Demand
If demand is variable or uncertain, EOQ gives the wrong answer.
Real-world application: use EOQ as a starting point, then add
safety stock for demand variability and lead time uncertainty.
Total cost curve is flat near EOQ — ±20% from optimal Q changes
total cost by only ~2%.

Phase 1: Input Validation

Determine: D (annual demand in units), S (fixed cost per order), H (holding cost per unit per year = unit cost × holding rate, typically 20-30% of unit value). Gate: All costs positive, demand estimate reasonable.

Phase 2: Core Algorithm

1. EOQ = √(2 × D × S / H)
2. Number of orders per year = D / EOQ
3. Reorder point = d × L (daily demand × lead time in days)
4. Total annual cost = (D/Q × S) + (Q/2 × H) at Q = EOQ

Phase 3: Verification

Check: ordering cost component ≈ holding cost component (they're equal at EOQ). Total cost is at minimum. Gate: Ordering cost ≈ holding cost (±5%).

Phase 4: Output

Return EOQ with cost breakdown and reorder point.

Output Format

{
  "eoq": 500,
  "orders_per_year": 20,
  "reorder_point": 150,
  "annual_cost": {"ordering": 2000, "holding": 2000, "total": 4000},
  "metadata": {"demand": 10000, "order_cost": 100, "holding_cost": 4.0}
}

Examples

Sample I/O

Input: D=10,000 units/year, S=$100/order, H=$4/unit/year Expected: EOQ = √(2×10000×100/4) = √500000 = 707 units

Edge Cases

Input	Expected	Why
Very high S, low H	Large EOQ, few orders	Minimize expensive ordering
Very low S, high H	Small EOQ, frequent orders	Minimize expensive holding
D = 0	EOQ = 0, no ordering	No demand, no orders needed

Gotchas

• Holding cost underestimation: H should include: capital cost, storage, insurance, obsolescence, handling. Companies often only count warehouse rent, understating true H.
• Flat cost curve: Total cost is insensitive near EOQ. Rounding EOQ to a convenient number (full pallet, container) costs very little.
• Quantity discounts: Price breaks at certain quantities may make it cheaper to order MORE than EOQ. Compare total cost at EOQ vs discount breakpoints.
• Lead time variability: EOQ doesn't address when to order, only how much. Add safety stock: SS = z × σ_demand × √(lead time).
• Multi-item coordination: When multiple items share ordering costs (same supplier), use joint replenishment models, not individual EOQs.

Scripts

Script	Description	Usage
scripts/eoq.py	Compute Economic Order Quantity and cost breakdown	python scripts/eoq.py --help

Run python scripts/eoq.py --verify to execute built-in sanity tests.

References

• For EOQ with quantity discounts, see references/eoq-discounts.md
• For safety stock calculation, see algo-sc-safety-stock