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Claude Code: EOQ, 报童模型，(s,S) 策略的深度研究⚓︎

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本内容完全生成自Claude Code的深度研究Skills。来源链接

经过少量人工核查和来自3个不同AI的辅助核查，修复了 2.4 离散需求下期望成本边际分析的错误。

  • 补充了 3.4 最优参数的适用范围 (by Gemini)

  • 补充了 1.4 EOQ 扩展模型的变量范围 (by Gemini)

注意本文依然可能存在遗漏或者错误，如发现请及时联系作者（笑笑）。

原内容参考笔记历史记录。

Executive Summary⚓︎

This report provides complete mathematical derivations and comparative analysis of three fundamental inventory models:

Model Type Key Decision Optimal Formula
EOQ Deterministic, Multi-period Order Quantity Q \(Q^* = \sqrt{\frac{2DK}{h}}\)
Newsvendor Stochastic, Single-period Stock Level Q \(F(Q^*) = \frac{C_u}{C_u + C_o}\)
(s,S) Policy Stochastic, Multi-period Reorder Point s, Order-up-to S \(G(s) = G(S) + K\)

1. EOQ Model and Extensions⚓︎

1.1 Basic Model Assumptions⚓︎

  • Demand: Constant, known, uniform over time (deterministic rate \(D\) units/year)
  • Replenishment: Instantaneous (entire order arrives at once)
  • Shortages: Not allowed (basic model)
  • Costs:
  • \(K\) = Fixed ordering cost per order
  • \(h\) = Holding cost per unit per year
  • \(C\) = Unit purchase cost

1.2 Complete Derivation⚓︎

Step 1: Define total annual cost function

\[TC(Q) = \frac{D}{Q}K + \frac{Q}{2}h + DC\]

Step 2: Take first derivative with respect to \(Q\)

\[\frac{dTC}{dQ} = -\frac{DK}{Q^2} + \frac{h}{2}\]

Step 3: Set derivative equal to zero for optimality

\[-\frac{DK}{Q^2} + \frac{h}{2} = 0\]

Step 4: Solve for \(Q^*\)

\[Q^* = \sqrt{\frac{2DK}{h}}\]

Step 5: Verify second-order condition

\[\frac{d^2TC}{dQ^2} = \frac{2DK}{Q^3} > 0 \quad \text{for } Q > 0\]

1.3 Key Results⚓︎

Metric Formula
Optimal Order Quantity \(Q^* = \sqrt{\frac{2DK}{h}}\)
Optimal Cycle Time \(t^* = \frac{Q^*}{D} = \sqrt{\frac{2K}{Dh}}\)
Minimum Total Cost \(TC^* = \sqrt{2DKh} + DC\)
Number of Orders per Year \(n = \frac{D}{Q^*} = \sqrt{\frac{Dh}{2K}}\)

1.4 Model Extensions⚓︎

Model II: EOQ with Backorders⚓︎

\[Q^* = \sqrt{\frac{2DK}{h}} \cdot \sqrt{\frac{h+p}{p}}, \quad b^* = Q^* \cdot \frac{h}{h+p}\]

Where \(p\) = shortage cost per unit per year.

Where \(b^*\) = maximum backordered quantity

Model III: Economic Production Quantity (EPQ)⚓︎

\[Q^* = \sqrt{\frac{2DK}{h}} \cdot \sqrt{\frac{P}{P-D}}\]

Where \(P\) = production rate (\(P > D\)).

Model V: Quantity Discounts⚓︎

Requires iterative algorithm comparing total costs at each price break point.

2. Newsvendor Model (Newsboy Problem)⚓︎

2.1 Model Assumptions⚓︎

  • Demand: Random, follows known probability distribution \(F(x)\)
  • Replenishment: Single ordering opportunity before selling season
  • Shortages: Lost sales (not backordered)
  • Costs:
  • \(c\) = Unit purchase cost
  • \(p\) = Unit selling price (\(p > c\))
  • \(s\) = Unit salvage value (\(s < c\))
  • \(C_u = p - c\) = Underage cost (lost profit per unit)
  • \(C_o = c - s\) = Overage cost (loss per unsold unit)

2.2 Complete Derivation (Continuous Demand)⚓︎

Step 1: Expected profit function

\[E[\pi(Q)] = p \cdot E[\min(Q,D)] + s \cdot E[\max(Q-D,0)] - cQ\]

Step 2: Express using integrals

\[E[\pi(Q)] = p\int_0^Q x f(x)dx + p\int_Q^\infty Q f(x)dx + s\int_0^Q (Q-x)f(x)dx - cQ\]

Step 3: Simplify

\[E[\pi(Q)] = (p-c)Q - (p-s)\int_0^Q F(x)dx\]

Step 4: First-order condition

\[\frac{dE[\pi]}{dQ} = (p-c) - (p-s)F(Q) = 0\]

Step 5: Solve for \(Q^*\)

\[F(Q^*) = \frac{p-c}{p-s} = \frac{C_u}{C_u + C_o}\]

Step 6: Second-order condition

\[\frac{d^2E[\pi]}{dQ^2} = -(p-s)f(Q) < 0 \quad \text{(confirms maximum)}\]

2.3 Critical Fractile Formula⚓︎

\[\boxed{Q^* = F^{-1}\left(\frac{C_u}{C_u + C_o}\right)}\]
Demand Distribution Optimal \(Q^*\)
Normal(\(\mu, \sigma^2\)) \(\mu + z_\alpha \sigma\) where \(z_\alpha = \Phi^{-1}(CR)\)
Uniform(\(a, b\)) \(a + CR \cdot (b-a)\)
Exponential(\(\lambda\)) \(-\frac{\ln(1-CR)}{\lambda}\)

2.4 Discrete Demand Case (Model VI)⚓︎

Using marginal analysis for expected cost:

\[\Delta C(Q) = C(Q+1) - C(Q) = (C_o + C_u)\sum_{r=0}^Q P(r) - C_u\]

Optimal \(Q^*\) satisfies \(\Delta C(Q^*) \ge 0\) and \(\Delta C(Q^*-1) < 0\), which yields:

\[F(Q^*-1) < \frac{C_u}{C_u + C_o} \leq F(Q^*)\]

3. (s,S) Inventory Policy⚓︎

3.1 Model Assumptions⚓︎

  • Demand: Stochastic, i.i.d. across periods
  • Replenishment: Instantaneous or with lead time \(L\)
  • Shortages: Backordered with penalty cost
  • Costs:
  • \(K\) = Fixed ordering cost
  • \(h\) = Holding cost per unit per period
  • \(p\) = Shortage cost per unit per period
  • \(c\) = Unit purchase cost

3.2 Policy Definition⚓︎

\[(s,S) \text{ Policy: } \begin{cases} \text{If } I < s: & \text{Order } S - I \\ \text{If } I \geq s: & \text{Do not order} \end{cases}\]

Where \(I\) = current inventory level.

3.3 Mathematical Foundation: K-Convexity⚓︎

Definition: A function \(g(x)\) is K-convex if for any \(x_1, x_2\) and \(\lambda \in [0,1]\):

\[g(\lambda x_1 + (1-\lambda)x_2) \leq \lambda g(x_1) + (1-\lambda)g(x_2) + K(1-\lambda)\]

Scarf's Theorem (1960): Under general conditions, the value function in the dynamic inventory problem is K-convex, and an (s,S) policy is optimal.

3.4 Characterization of Optimal Parameters⚓︎

Let \(G(y) = h \cdot E[(y-D)^+] + p \cdot E[(y-D)^-]\) be the single-period expected cost.

Optimal \(S^*\): Minimizes \(G(y)\)

\[\frac{dG}{dy} = 0 \implies F(S^*) = \frac{p}{h+p}\]

Note: This strictly applies to the single-period problem; for multi-period, it serves as a myopic approximation.

Optimal \(s^*\): Satisfies the indifference condition

\[G(s^*) = G(S^*) + K\]

3.5 Solution Method⚓︎

  1. Compute critical ratio: \(CR = \frac{p}{h+p}\)
  2. Find \(S^* = F^{-1}(CR)\)
  3. Solve \(G(s) = G(S^*) + K\) numerically for \(s\)

For normal demand with lead time \(L\):

\[S^* = \mu_L + z \cdot \sigma_L \quad \text{where } z = \Phi^{-1}(CR)\]

4. Comparative Analysis⚓︎

4.1 Decision Framework⚓︎

Use Model When
EOQ: Demand is stable and predictable; holding and ordering costs dominate
Newsvendor: Single selling opportunity; high demand uncertainty; perishable goods
(s,S): Stochastic demand over multiple periods; significant fixed ordering costs

4.2 Assumptions Comparison⚓︎

Aspect EOQ Newsvendor (s,S)
Demand Deterministic Stochastic Stochastic
Time Horizon Infinite Single Period Infinite
Review Continuous Single Continuous/Periodic
Fixed Order Cost Yes N/A Yes
Key Trade-off Holding vs. Ordering Overage vs. Underage Three-way balance

4.3 Real-World Applications⚓︎

Industry EOQ Newsvendor (s,S)
Retail Staple goods Fashion, seasonal items High-value electronics
Healthcare Routine supplies Vaccines (expiry) Critical medications
Manufacturing Component production Customized products MRO items

5. Key Insights⚓︎

EOQ Model⚓︎

  • The square-root relationship means quadrupling demand only doubles optimal order quantity
  • Total ordering cost equals total holding cost at the EOQ (in basic model)
  • Cost curve is flat near optimum - model is robust to parameter errors

Newsvendor Model⚓︎

  • The critical ratio \(\frac{C_u}{C_u+C_o}\) represents the optimal service level
  • Higher underage cost leads to higher order quantities
  • When \(C_u = C_o\), optimal service level is 50%

(s,S) Policy⚓︎

  • K-convexity is the mathematical property guaranteeing (s,S) optimality
  • The gap \(S-s\) increases with setup cost \(K\)
  • As \(K \to 0\), (s,S) converges to base-stock policy

6. References⚓︎

Foundational Papers⚓︎

  1. Harris, F.W. (1913). "How Many Parts to Make at Once". Factory, The Magazine of Management, 10(2), 135-136.
  2. Arrow, K.J., Harris, T., & Marschak, J. (1951). "Optimal Inventory Policy". Econometrica, 19(3), 250-272.
  3. Scarf, H. (1960). "The Optimality of (s,S) Policies in the Dynamic Inventory Problem". In Mathematical Methods in the Social Sciences, Stanford University Press.

Textbooks⚓︎

  1. Zipkin, P.H. (2000). Foundations of Inventory Management. McGraw-Hill.
  2. Silver, E.A., Pyke, D.F., & Peterson, R. (1998). Inventory Management and Production Planning and Scheduling. Wiley.
  3. Nahmias, S. & Olsen, T.L. (2015). Production and Operations Analysis. Waveland Press.

Report generated from deep research conducted on 2026-03-10