8.2 Inventory Control Models: EOQ, Reorder Point & Safety Stock

8.2 Inventory Control Models: EOQ, Reorder Point & Safety Stock

To manage inventory efficiently, organizations utilize mathematical control models that answer two fundamental questions: How much to order? and When to order? The CPSM Exam 2 thoroughly tests candidate fluency in deriving and executing calculations for the Economic Order Quantity (EOQ), Reorder Point (ROP), and Safety Stock.

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The Economic Order Quantity (EOQ) Model

The Economic Order Quantity (EOQ) is a classic continuous-review lot-sizing model designed to identify the order quantity ($Q$) that minimizes the total annual cost of managing inventory.

1. Underlying Model Assumptions

• Demand ($D$) is deterministic, continuous, and known with certainty.
• Lead time ($L$) is constant and known.
• Replenishment occurs instantaneously in a single batch delivery.
• No stockouts or shortages are permitted.
• Purchase price ($C$) per unit is fixed and independent of order quantity (no quantity discounts).
• Ordering cost ($S$) per order is fixed regardless of order quantity.
• Annual carrying cost rate ($H$) per unit per year is constant.

2. Mathematical Derivation of EOQ

Total Annual Inventory Cost ($TC$) consists of Annual Ordering Cost plus Annual Carrying Cost: $\text{Annual Ordering Cost} = \left( \frac{D}{Q} \right) \times S$ $\text{Annual Carrying Cost} = \left( \frac{Q}{2} \right) \times H$ $\text{Total Annual Cost } (TC) = \left( \frac{D}{Q} \right) S + \left( \frac{Q}{2} \right) H$

To find the minimum cost point, take the first derivative of $TC$ with respect to $Q$ and set it equal to zero: $\frac{d(TC)}{dQ} = -\frac{D \cdot S}{Q^2} + \frac{H}{2} = 0 \implies \frac{D \cdot S}{Q^2} = \frac{H}{2} \implies Q^2 = \frac{2DS}{H}$

Taking the square root yields the classic EOQ formula: $EOQ = \sqrt{\frac{2DS}{H}}$ where:

• $D$ = Annual demand in units
• $S$ = Fixed ordering cost per purchase order ($/order)
• $H$ = Annual holding/carrying cost per unit ($/unit/year), where $H = C \times h$ ($C$ = unit cost, $h$ = holding cost %)

3. The EOQ Trade-off Principle

At the optimal EOQ quantity: $\text{Annual Ordering Cost} = \text{Annual Carrying Cost}$ If order quantity $Q > EOQ$, carrying costs dominate. If $Q < EOQ$, ordering costs dominate.

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Reorder Point (ROP) Mechanics

While EOQ determines how much to order, the Reorder Point (ROP) determines when to place the order.

1. Continuous Review ($Q$-System) vs. Periodic Review ($P$-System)

• Continuous Review ($Q$-System): Inventory status is continuously tracked. Whenever inventory drops to the predefined ROP, a fixed quantity ($Q = EOQ$) is ordered.
• Periodic Review ($P$-System): Inventory is counted at fixed time intervals (e.g., every 30 days). Variable quantities are ordered to restore inventory to a target target level ($T$).

2. Deterministic Reorder Point Formula

Under constant daily demand ($d$) and constant supplier lead time ($L$ days): $ROP = d \times L$ where $d = \frac{D}{\text{Operating Days per Year}}$.

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Safety Stock (SS) Determination under Uncertainty

In real-world operations, daily demand and vendor lead times fluctuate. To prevent stockouts during lead time, organizations carry Safety Stock (SS).

1. Variable Demand with Constant Lead Time

When daily demand follows a normal distribution with standard deviation $\sigma_d$, the standard deviation of demand during lead time ($\sigma_{DLT}$) is: $\sigma_{DLT} = \sqrt{L} \times \sigma_d$

Safety Stock is calculated as: $SS = Z \times \sigma_{DLT} = Z \times \sigma_d \times \sqrt{L}$ where $Z$ is the standard normal z-score corresponding to the desired Cycle Service Level (CSL):

• 90% Service Level: $Z = 1.282$
• 95% Service Level: $Z = 1.645$
• 98% Service Level: $Z = 2.050$
• 99% Service Level: $Z = 2.326$

2. Stochastic Reorder Point Formula

When safety stock is maintained, the reorder point formula becomes: $ROP = (d \times L) + SS$

3. Combined Demand and Lead Time Variability

When both daily demand (with standard deviation $\sigma_d$) and lead time (with average $L$ and standard deviation $\sigma_L$) vary independently: $\sigma_{DLT} = \sqrt{L \cdot \sigma_d^2 + d^2 \cdot \sigma_L^2}$ $SS = Z \times \sqrt{L \cdot \sigma_d^2 + d^2 \cdot \sigma_L^2}$

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Step-by-Step Worked Numerical Calculation: EOQ, ROP, and Safety Stock

Scenario:

An industrial distributor sells hydraulic valves under the following operating conditions:

• Annual Demand ($D$): 12,000 units
• Operating Days per Year: 300 days ($d = 12,000 / 300 = 40$ units/day)
• Purchase Price ($C$): $50 per unit
• Ordering Cost ($S$): $150 per order
• Annual Carrying Cost Rate ($h$): 20% ($H = $50 \times 0.20 = $10$ per unit/year)
• Supplier Lead Time ($L$): 5 business days
• Daily Demand Standard Deviation ($\sigma_d$): 8 units
• Desired Cycle Service Level: 95% ($Z = 1.645$)

Step-by-Step Calculation:

1. Calculate Economic Order Quantity (EOQ): $EOQ = \sqrt{\frac{2 \times 12,000 \times 150}{10}} = \sqrt{\frac{3,600,000}{10}} = \sqrt{360,000} = 600 \text{ units}$

2. Calculate Number of Orders per Year: $\text{Orders per Year} = \frac{D}{EOQ} = \frac{12,000}{600} = 20 \text{ orders/year}$

3. Calculate Annual Ordering Cost and Carrying Cost: $\text{Annual Ordering Cost} = 20 \times \$150 = \$3,000$ $\text{Annual Carrying Cost} = \left( \frac{600}{2} \right) \times \$10 = 300 \times \$10 = \$3,000$ Verification: Ordering cost ($3,000) equals carrying cost ($3,000). Total annual inventory management cost = $6,000.

4. Calculate Expected Demand During Lead Time ($d \times L$): $\text{Expected Lead Time Demand} = 40 \text{ units/day} \times 5 \text{ days} = 200 \text{ units}$

5. Calculate Safety Stock ($SS$): $SS = Z \times \sigma_d \times \sqrt{L} = 1.645 \times 8 \times \sqrt{5} = 1.645 \times 8 \times 2.23607 = 29.42 \approx 30 \text{ units}$

6. Calculate Reorder Point (ROP): $ROP = (d \times L) + SS = 200 + 30 = 230 \text{ units}$

Operational Conclusion: The distributor should place a purchase order for 600 units whenever inventory on hand plus on order drops to 230 units.