01 — Notebook Information & Scope

• Domain: Operations & Supply Chain Systems
• Subject: Logistics & Supply Chain Management
• Pedagogical Lead: Prof. Manoj Dagaonkar
• Core Reference Models: Ford W. Harris Classical EOQ, Taft EPQ Model, Silver-Meal Heuristic, and Stochastic Safety Stock Normal Theory

02 — Learning Map

[Demand Characteristics]
   ├── Deterministic / Constant ──> [EOQ / EPQ Lot Sizing] ──> [Quantity Discount Optimization]
   └── Stochastic / Uncertain   ──> [Lead-Time Demand Variance] ──> [Safety Stock: Z * sigma_LTD]
                                                                        │
                                                                        ▼
                                                             [Newsvendor Critical Fractile]

03 — The Classical Economic Order Quantity (EOQ)

The Ford W. Harris (1913) model identifies the optimal order quantity that minimizes the sum of annual ordering costs and annual inventory holding costs under idealized deterministic conditions:

Fundamental Assumptions

1. Demand ($D$) is continuous, constant, and known with absolute certainty.
2. Replenishment lead time ($L$) is constant and known.
3. Inventory arrives in a single instantaneous lot ($Q$).
4. Unit purchase price ($C$) is fixed; zero quantity discounts.
5. No stockouts are permitted ($SS = 0$).

Total Annual Cost Equation

The annual total cost function $TC(Q)$ comprises purchase cost, ordering cost, and carrying cost:

Where:

• $D$ = Annual Demand (units/year)
• $S$ = Ordering cost per order ($/order)
• $H$ = Annual inventory holding cost per unit ($H = i \cdot C$, where $i$ is holding cost rate)
• $Q$ = Order quantity (units)

Key EOQ Metric Relationships:

• Optimal Annual Orders: $N^* = \frac{D}{Q^*} = \sqrt{\frac{DH}{2S}}$
• Optimal Cycle Time (Between Orders): $T^* = \frac{Q^*}{D} = \sqrt{\frac{2S}{DH}}$
• Minimum Total Annual Variable Cost: $TVC^* = \sqrt{2DSH}$

04 — Economic Production Quantity (EPQ / EBQ)

When replenishments are manufactured in-house rather than procured instantaneously from an external vendor, inventory accumulates at production rate $p$ while being depleted simultaneously by sales demand rate $d$ ($p > d$):

05 — Stochastic Inventory: Safety Stock Under Demand Uncertainty

In real-world commercial environments, customer demand fluctuates around a statistical mean $\mu_d$ with standard deviation $\sigma_d$.

Continuous Review System $(r, Q)$

A continuous review system continuously monitors inventory. Whenever inventory position (On-Hand + On-Order - Backorders) falls to or below the Reorder Point ($ROP$), a replenishment lot of fixed size $Q$ is dispatched.

Both Demand and Lead Time are Stochastic

When supplier lead time is also uncertain (mean $\bar{L}$, standard deviation $\sigma_L$):

06 — The Single-Period Newsvendor Problem

For perishable goods, high-fashion apparel, and seasonal items with a single purchasing decision and zero replenishment opportunity:

07 — Step-by-Step Worked Numerical Problem

Problem Statement

A manufacturing assembly plant operates 250 days per year.

• Annual demand for raw component K-9: $D = 10,000$ units.
• Cost per order: $S = $50$.
• Unit cost: $C = $20$.
• Annual inventory holding cost rate: $i = 20%$ per year ($H = 0.20 \times 20 = $4$ per unit/year).
• Daily demand is normally distributed with mean $\bar{d} = 40$ units/day and $\sigma_d = 8$ units/day.
• Replenishment lead time is constant: $L = 9$ business days.
• Desired Cycle Service Level ($CSL$): $95%$ ($Z = 1.645$).

Step 1: Calculate Economic Order Quantity ($EOQ$)

$Q^* = \sqrt{\frac{2 \times 10000 \times 50}{4}} = \sqrt{\frac{1000000}{4}} = \sqrt{250000} = 500\text{ units}$

Step 2: Calculate Number of Orders and Cycle Time

• Orders per year: $N^* = \frac{10000}{500} = 20\text{ orders/year}$
• Cycle time: $T^* = \frac{250\text{ days}}{20} = 12.5\text{ days}$

Step 3: Calculate Lead-Time Demand & Safety Stock

• Mean lead-time demand: $d_L = \bar{d} \times L = 40 \times 9 = 360\text{ units}$
• Standard deviation of lead-time demand: $\sigma_L = \sigma_d \times \sqrt{L} = 8 \times \sqrt{9} = 8 \times 3 = 24\text{ units}$
• Safety Stock: $SS = Z \times \sigma_L = 1.645 \times 24 = 39.48 \approx 40\text{ units}$

Step 4: Calculate Reorder Point ($ROP$)

$ROP = d_L + SS = 360 + 40 = 400\text{ units}$

Step 5: Total Annual Inventory Holding Cost (Cycle + Safety)

• Average Cycle Inventory: $\frac{Q^*}{2} = \frac{500}{2} = 250\text{ units}$
• Total Average Inventory: $250 + 40 = 290\text{ units}$
• Total Annual Inventory Cost: $Total = \left(\frac{10000}{500} \times 50\right) + (250 \times 4) + (40 \times 4) = 1000 + 1000 + 160 = \$2,160$

08 — Past Examination Questions & Model Solutions

09 — Active Recall Flashcards & Conceptual Quiz

10 — Knowledge Graph & Cross-References