The Economic Order Quantity (EOQ) model is a fundamental inventory-management technique used to determine the most economical quantity of inventory that a firm should order each time it replenishes its stock. The model was developed to establish an optimal balance between the costs of placing orders and the costs of holding inventory.
The core objective of EOQ is therefore to minimise the total annual inventory cost, particularly the combined ordering cost and carrying (holding) cost, while ensuring that the firm's inventory requirements are met. EOQ identifies the order quantity at which these relevant costs are balanced.
Core Objective of the EOQ Model
A business faces two major costs when deciding how much inventory to order:
- Ordering cost: The cost incurred every time an order is placed, such as purchasing administration, transportation arrangements, receiving, and inspection costs. If a firm places many small orders, the number of orders increases and total ordering cost rises.
- Holding or carrying cost: The cost of keeping inventory in stock, including storage, insurance, handling, deterioration, and the opportunity cost of capital invested in inventory. If a firm orders large quantities, its average inventory increases and therefore its holding cost rises.
EOQ determines the order quantity that minimises the sum of these two costs.
Mathematical Derivation of the EOQ Formula
Let:
- D = annual demand for the inventory item, in units
- Q = quantity ordered each time
- S = ordering cost per order
- H = annual holding cost per unit
- TC = relevant total annual inventory cost
Assume that demand is constant and inventory is replenished immediately. Each order of Q units creates an average inventory of Q/2 units because inventory continuously falls from Q to zero before being replenished.
1. Annual Ordering Cost
If annual demand is D units and each order contains Q units, the number of orders placed per year is:
Number of orders = D/Q
Since each order costs S, annual ordering cost is:
Ordering Cost = (D/Q)S
This shows that ordering cost decreases as Q increases because larger orders mean fewer orders are required.
2. Annual Holding Cost
With instantaneous replenishment and steady usage, inventory fluctuates between Q and zero. Therefore, average inventory is:
Average Inventory = Q/2
If the annual holding cost per unit is H, annual holding cost is:
Holding Cost = (Q/2)H
Thus, holding cost increases as Q increases.
3. Total Relevant Annual Inventory Cost
Ignoring the purchase cost, which remains constant when the unit purchase price does not depend on order quantity, total relevant inventory cost is:
TC = (D/Q)S + (Q/2)H
To find the order quantity that minimises total cost, differentiate TC with respect to Q:
dTC/dQ = -DS/Q² + H/2
For the minimum-cost quantity, set the first derivative equal to zero:
-DS/Q² + H/2 = 0
Therefore:
H/2 = DS/Q²
Multiplying both sides by Q²:
HQ² = 2DS
Dividing by H:
Q² = 2DS/H
Taking the positive square root:
EOQ = √(2DS/H)
Thus, the basic EOQ formula is:
EOQ = √(2DS/H)
The second derivative is:
d²TC/dQ² = 2DS/Q³
Since D, S, and Q are positive, this expression is positive. Therefore, the value obtained from the first-order condition represents a minimum, confirming that EOQ minimises the relevant inventory costs.
Underlying Assumptions of the Basic EOQ Model
The basic EOQ model depends on several simplifying assumptions:
1. Constant and known demand: Annual demand is assumed to be certain and constant throughout the period.
2. Constant ordering cost: The cost of placing one order, S, remains unchanged regardless of the quantity ordered.
3. Constant holding cost: The annual carrying cost per unit, H, is assumed to remain constant.
4. Instantaneous replenishment: The entire order is assumed to arrive at once, so there is no gradual delivery.
5. No stock-outs: Inventory is replenished before it reaches a level that would cause shortages. Therefore, stock-out costs are ignored.
6. Constant lead time: The time between placing an order and receiving it is assumed to be known and constant.
7. No quantity discounts: The unit purchase price does not change with the size of the order. Consequently, purchase cost does not affect the EOQ calculation.
8. Single-product analysis: The basic model normally considers one inventory item independently and assumes there are no constraints on storage space or purchasing funds.
9. No deterioration or obsolescence: Inventory is assumed to remain usable throughout the holding period.
Conclusion
The EOQ model provides a systematic method for determining the optimal order quantity by balancing ordering and holding costs. Its central objective is to minimise total relevant inventory costs while maintaining an adequate supply of stock. By equating the economic effects of ordering too frequently with those of holding excessive inventory, the model produces the formula EOQ = √(2DS/H). Although its assumptions are restrictive and real-world inventory systems may require modifications, EOQ remains an important foundation for inventory-control decisions.
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