Economic Order Quantity (EOQ) Model

Introduction

The EOQ model is a fundamental inventory management technique used to determine the optimal order quantity that minimizes the total inventory cost. This includes ordering costs and holding costs.

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Objective

To find the quantity $Q$ that minimizes the total cost (TC), which is the sum of:

• Ordering cost: Cost incurred every time an order is placed
• Holding cost: Cost to hold one unit of inventory for a year

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Assumptions

• Constant and known demand ($D$ units/year)
• Fixed ordering cost per order ($S$)
• Constant holding cost per unit per year ($H$)
• No stockouts or shortages
• Instantaneous replenishment

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Total Cost (TC) Function

The total cost is given by:

$TC(Q) = \frac{D}{Q} \cdot S + \frac{Q}{2} \cdot H$

Where:

• $D$: Annual demand
• $S$: Ordering cost per order
• $H$: Holding cost per unit per year
• $Q$: Order quantity (decision variable)

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EOQ Formula

To minimize $TC(Q)$, take the derivative of $TC$ with respect to $Q$ and set it to zero:

$$\frac{dTC}{dQ} = -\frac{D \cdot S}{Q^2} + \frac{H}{2} = 0$$

Solving gives:

$$Q^* = \sqrt{\frac{2DS}{H}}$$

Where:

• $Q^*$: Optimal order quantity (EOQ)

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Example

Given:

• $D = 10{,}000$ units/year
• $S = 100$ per order
• $H = 2$ per unit per year

Then:

$$Q^* = \sqrt{\frac{2 \cdot 10{,}000 \cdot 100}{2}} = \sqrt{1{,}000{,}000} = 1000$$

So, the optimal order quantity is 1,000 units.

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Interpretation

• Fewer orders → Higher holding cost
• More orders → Higher ordering cost
• EOQ balances both to minimize total cost

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Graph Interpretation

A typical EOQ cost graph shows:

• $TC(Q)$: U-shaped total cost curve
• Minimum point corresponds to $Q^*$
• The intersection of ordering and holding cost curves

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Summary

Symbol	Meaning
$D$	Annual demand
$S$	Cost per order
$H$	Holding cost per unit/year
$Q$	Order quantity
$Q^*$	EOQ (Optimal order quantity)

EOQ is a key tool for reducing inventory costs in supply chain and operations management.