algo-sc-eoq

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Economic Order Quantity (EOQ)

经济订货批量（EOQ）

Overview

概述

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.

EOQ用于确定能最小化总库存成本（订货成本+持有成本）的订货量。公式为：EOQ = √(2DS/H)，其中D=年需求量，S=每次订货的订货成本，H=单位产品年持有成本。该模型假设需求恒定且补货瞬时完成。

When to Use

适用场景

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

触发条件：

为库存补货设定标准订货量
平衡订货频率与仓储成本
在调整安全库存前进行基准计算

不适用场景：

需求高度不确定时（使用报童模型）
产品易腐、保质期短
数量折扣显著改变成本结构时

Algorithm

算法

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%.

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

阶段1：输入验证

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.

确定：D（年需求量，单位为件），S（每次订货的固定成本），H（单位产品年持有成本=单位成本×持有费率，通常为单位价值的20-30%）。 **校验门限：**所有成本为正数，需求估算合理。

Phase 2: Core Algorithm

阶段2：核心算法

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

EOQ = √(2 × D × S / H)
年订货次数 = D / EOQ
再订货点 = d × L（日需求量×提前期天数）
年总成本 = (D/Q × S) + (Q/2 × H)，其中Q = EOQ

Phase 3: Verification

阶段3：验证

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

检查：订货成本部分≈持有成本部分（在EOQ点两者相等）。总成本处于最小值。 **校验门限：**订货成本≈持有成本（误差±5%）。

Phase 4: Output

阶段4：输出

Return EOQ with cost breakdown and reorder point.

返回EOQ及成本明细、再订货点。

Output Format

输出格式

json

{
  "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}
}

json

{
  "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

示例

Sample I/O

输入输出示例

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

**输入：**D=10,000件/年，S=100美元/次订货，H=4美元/件/年 **预期结果：**EOQ = √(2×10000×100/4) = √500000 = 707件

Edge Cases

边缘情况

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

输入	预期结果	原因
极高S，低H	大EOQ，订货次数少	最小化高昂的订货成本
极低S，高H	小EOQ，订货频繁	最小化高昂的持有成本
D = 0	EOQ = 0，无需订货	无需求，无需订货

Gotchas

注意事项

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.

**持有成本低估：**H应包含：资金成本、仓储费、保险费、损耗费、搬运费。企业常仅计算仓库租金，从而低估实际H。
**成本曲线平缓：**EOQ附近总成本变化不敏感。将EOQ取整为方便的数值（如整托盘、整集装箱），成本几乎无变化。
**数量折扣：**当达到特定订货量可享受价格折扣时，可能订比EOQ更多的货更划算。需比较EOQ点与折扣临界点的总成本。
**提前期波动：**EOQ仅解决订多少的问题，不解决何时订货。需添加安全库存：SS = z × σ_demand × √(提前期)。
**多产品协同：**当多个产品共享订货成本（同一供应商）时，应使用联合补货模型，而非单独计算各产品的EOQ。

Scripts

脚本

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.

脚本	描述	使用方法
`scripts/eoq.py`	计算经济订货批量及成本明细	`python scripts/eoq.py --help`

运行

python scripts/eoq.py --verify

可执行内置的合理性测试。

References

参考资料

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

带数量折扣的EOQ，请参阅
```
references/eoq-discounts.md
```
安全库存计算，请参阅algo-sc-safety-stock