Economic Lot Size Model

Total cost

$$C_t=K+\frac{hTQ}{2}$$

Average inventory level

$$\frac Q2$$

Cycle time

$$T=\frac{Q}{D}$$

Average total cost per unit time

$$\frac{C_t}{T}=\frac KT+\frac{hQ}{2}=\frac{KD}{Q}+\frac{hQ}{2}$$

Economic Order Quantity (EOQ)

$$Q=\arg\min_Q \frac{C_t}{T}=\arg\min_Q \frac{KD}{Q}+\frac{hQ}{2}=\sqrt{\frac{2KD}{h}}$$

Single Period Models

• Fixed cost $100,000
• Variable cost $80 per unit
• Sale price $125
• Salvage price $20

Scenario I

• Produce 10,000 units
• Sell 12,000 units (Demand)
• Probability 27%

125(10,000)-80(10,000)-100,000=\350,000$

Scenarios II

• Produce 10,000 units
• Sell 8,000 units
• Probability 11%

125(8,000)+20(2,000)-80(10,000)-100,000=\140,000$

Relationship between Optimal Quantity and Average Demand

• Marginal Profit > Marginal Cost => Optimal Quantity > Average Demand
• Marginal Profit < Marginal Cost => Optimal Quantity < Average Demand

Risk-Reward Tradeoffs

• Minimal Ordering Quantity (MOQ)
• Minimal Packing Quantity (MPQ)

Initial Inventory

Trade-off between

• Using on-hand inventory to meet demand and avoid paying fixed production cost: need sufficient inventory stock
• Paying the fixed cost of production and avoid lose of sales

Multiple Order Opportunities

Continuous Review Policy ((Q,R) Policy)

• Average daily demand, $AVG$
• Standard deviation of daily demand, $STD$
• Replenishment lead time, $L$
• Holding cost, $h$
• Service level, $\alpha$

Average demand during lead time

$$L\times AVG$$

Safety stock

$$z\times STD\times\sqrt L$$

Reorder level

$$R=L\times AVG + z\times STD\times \sqrt L$$

Average Inventory

$$\frac Q2+z\times STD\times\sqrt{L}$$

Order Quantity

$$Q=\sqrt{\frac{2K\times AVG}{h}}$$

Approximation from EOQ, $Q^{\star}=\sqrt{\frac{2KD}{h}}$

Periodic Review Policy

Short Intervals ((s,S) Policy)

Same as (Q, R) Policy

Longer Intervals (Base-stock level policy)

低价商品 (Nails and Screw)

• Average daily demand, $AVG$
• Standard deviation of daily demand, $STD$
• Lead time $L$
• Length of the review period $r$
• Service level, $\alpha$

Average demand during an interval of $r+L$

$$(r+L)\times AVG$$

Base-stock level

$$(r+L)\times AVG+z\times STD\times\sqrt{r+L}$$

Average Inventory

$$r\times AVG+ \mathrm{safety\ stock}$$

Make-to-Order Demo

Global Optimization

One Entity

• Sale price $125
• Salvage value $20
• Fixed cost $0
• Variable cost $35

Marginal profit $90, marginal cost $15

16,000 => $1,014,500

2 Stage Example

2 Stages (2 entities)

• Buyer (retailer)
• Supplier (manufacturer)

Buyer Information

• Sale price $125
• Wholesale price $80
• Salvage price $20

Marginal profit $45, marginal cost $60

12,000 => $470,700

Supplier Information

• Fixed cost $100,000
• Variable cost $35 per unit

$440,000

Buy-Back Contract

Buyer Information

• Buy-back price $55

Marginal profit $45, marginal cost $25

14,000 => $513,800

Supplier Information

$471,900

Revenue-Sharing Contract

Buyer Information

• Wholesale price $60
• 15% revenue to Supplier

14,000 => $504,325

Supplier Information

$481,375

Make-to-Stock Demo

Global Optimization

One Entity

• Sale price $125
• Salvage price $20
• Fixed cost $0
• Variable cost $55

Marginal profit $70, marginal cost $35

14,000 => $705,700

2-Stage Example

2 Stage

• Buyer (distributor)
• Supplier (manufacturer)

Buyer Information

• Sale price $125
• Wholesale price $80

$510,300

Supplier Information

• Fixed cost $100,000
• Variable cost $55 per unit
• Salvage price $20

Marginal profit $25, marginal cost $60

12,000 => $160,400

Pay-Back Contract

Buyer Information

$525,420

Supplier Information

• Pay-Back price $18

Marginal profit $25, marginal cost $17

14,000 => $180,280

Cost-Sharing Contract

Buyer Information

$523,320

Supplier Information

• Wholesale price $62

• Return 33% of the cost

14,000 => $182.380