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Economic Order Quantity (EOQ): Formula, Example, Assumptions, Advantages and Limitations
Inventory management is about maintaining the right amount of stock at the right time and at a reasonable cost. If a business orders inventory too frequently, its ordering costs may become high. If it orders very large quantities, more money is tied up in inventory and holding costs may increase.
Economic Order Quantity (EOQ) provides a quantitative way to balance these two important costs. It determines the order quantity that minimizes the relevant annual ordering and holding costs under the assumptions of the basic EOQ model.
EOQ is one of the fundamental deterministic inventory models studied in Operations Research and inventory management.
What is Economic Order Quantity (EOQ)?
Economic Order Quantity (EOQ) is the order quantity that minimizes the total relevant cost of ordering and holding inventory, under the assumptions of the basic EOQ model.
In simple words:
EOQ tells a business how many units it should order at one time so that the combined ordering and holding costs are minimized.
The basic idea is simple:
- If the order quantity is very small, the business has to place orders more frequently. Therefore, annual ordering cost increases.
- If the order quantity is very large, the business keeps more inventory on average. Therefore, annual holding cost increases.
- EOQ finds the order quantity that gives the minimum total of these two relevant costs.
The classical EOQ model is associated with Ford W. Harris, who presented the well-known model in 1913.
The Basic Idea Behind EOQ
Suppose a company needs a certain number of units every year.
It has two major inventory-related costs.
Ordering Cost
Ordering cost is the cost incurred each time an order is placed.
Examples include:
- purchase order processing
- administrative work
- communication and documentation
- receiving and inspection activities
- other costs that arise because an order is placed
If the company places many small orders, the number of orders increases and the annual ordering cost becomes higher.
Holding or Carrying Cost
Holding cost is the cost of keeping inventory in stock.
It may include:
- cost of capital tied up in inventory
- storage and warehouse costs
- insurance
- deterioration or spoilage
- handling and other inventory-related costs
If the company orders large quantities, its average inventory becomes higher and holding cost increases.
Therefore, inventory management involves a trade-off:
Small orders → higher ordering cost but lower holding cost
Large orders → lower ordering cost but higher holding cost
EOQ → the quantity that minimizes the combined relevant cost.
EOQ Formula
The basic EOQ formula is:
\[ EOQ=\sqrt{\frac{2DS}{H}} \]
Where:
- \(D\) = annual demand in units
- \(S\) = ordering cost per order
- \(H\) = annual holding cost per unit
Therefore:
\[ \boxed{EOQ=\sqrt{\frac{2DS}{H}}} \]
The answer is expressed in units.
Meaning of the Variables in the EOQ Formula
| Symbol | Meaning | Typical Unit |
|---|---|---|
| \(D\) | Annual demand | units/year |
| \(S\) | Ordering cost per order | currency/order |
| \(H\) | Annual holding cost per unit | currency/unit/year |
| \(Q\) | Order quantity | units |
| \(EOQ\) | Economic Order Quantity | units |
Different Operations Research books may use different symbols. For example, ordering cost may be represented by \(C_o\) or \(C_3\), while holding cost may be represented by \(C_h\), \(C_1\), or \(H\).
The symbols may change, but the underlying model remains the same.
Why Does the EOQ Formula Work?
The basic EOQ model considers two relevant costs:
Annual Ordering Cost
If annual demand is \(D\) and each order contains \(Q\) units, then the number of orders per year is:
\[ =\frac{D}{Q} \]
If the cost of placing one order is \(S\), then:
\[ \text{Annual Ordering Cost}=\frac{D}{Q}S \]
Annual Holding Cost
Under the basic EOQ assumptions, inventory is replenished in such a way that inventory falls from \(Q\) to zero between replenishments.
Therefore, average inventory is:
\[ =\frac{Q}{2} \]
If the annual holding cost per unit is \(H\), then:
\[ \text{Annual Holding Cost}=\frac{Q}{2}H \]
Therefore, the total relevant annual inventory cost is:
\[ TC(Q)=\frac{D}{Q}S+\frac{Q}{2}H \]
The EOQ is the value of \(Q\) that minimizes this total relevant cost.
Thus:
\[ \boxed{EOQ=\sqrt{\frac{2DS}{H}}} \]
Mathematical Derivation of EOQ / EOQ derivation with calculus with step-by-step
The EOQ formula can be derived by finding the order quantity that minimizes the total relevant inventory cost.
Let:
- \(D\) = annual demand in units
- \(Q\) = order quantity in units
- \(S\) = ordering cost per order
- \(H\) = annual holding cost per unit
We consider the basic EOQ model, where purchase price is constant and shortages are not allowed.
Step 1: Calculate the Number of Orders per Year
If annual demand is \(D\) units and each order contains \(Q\) units, then the number of orders placed per year is:
\[ N=\frac{D}{Q} \]
Step 2: Calculate Annual Ordering Cost
Each order costs \(S\).
Therefore:
\[ \text{Annual Ordering Cost}=\frac{D}{Q}S \]
Step 3: Calculate Average Inventory
Under the basic EOQ model, inventory decreases from \(Q\) units to zero before the next replenishment arrives.
Therefore:
\[ \text{Average Inventory}=\frac{Q+0}{2} \]
Hence:
\[ \text{Average Inventory}=\frac{Q}{2} \]
Step 4: Calculate Annual Holding Cost
If the annual holding cost per unit is \(H\), then:
\[ \text{Annual Holding Cost}=\frac{Q}{2}H \]
Step 5: Write the Total Relevant Cost Function
The total relevant annual inventory cost consists of ordering cost and holding cost:
\[ TC(Q)=\frac{D}{Q}S+\frac{Q}{2}H \]
This is the cost function that we want to minimize.
Step 6: Differentiate the Total Cost Function
Differentiate \(TC(Q)\) with respect to \(Q\):
\[ \frac{dTC}{dQ}=-\frac{DS}{Q^2}+\frac{H}{2} \]
At the minimum-cost point, the first derivative must be zero:
\[ \frac{dTC}{dQ}=0 \]
Therefore:
\[ -\frac{DS}{Q^2}+\frac{H}{2}=0 \]
Step 7: Rearrange the Equation
Move the first term to the other side:
\[ \frac{H}{2}=\frac{DS}{Q^2} \]
Multiply both sides by \(2Q^2\):
\[ HQ^2=2DS \]
Therefore:
\[ Q^2=\frac{2DS}{H} \]
Taking the positive square root:
\[ Q=\sqrt{\frac{2DS}{H}} \]
Since \(Q\) represents an order quantity, we take the positive value.
Therefore:
\[ \boxed{EOQ=\sqrt{\frac{2DS}{H}}} \]
Step 8: Verify That the Result Is a Minimum
The second derivative is:
\[ \frac{d^2TC}{dQ^2}=\frac{2DS}{Q^3} \]
For positive values of \(D\), \(S\), and \(Q\):
\[ \frac{2DS}{Q^3}>0 \]
Therefore, the total cost function is convex at the solution, and the calculated EOQ gives a minimum total relevant cost.
Hence:
\[ \boxed{EOQ=\sqrt{\frac{2DS}{H}}} \]
is the optimal order quantity for the basic EOQ model.
Why are Ordering Cost and Holding Cost Equal at EOQ?
The equality can also be obtained directly from the first-order condition.
At EOQ:
\[ -\frac{DS}{Q^2}+\frac{H}{2}=0 \]
Therefore:
\[ \frac{DS}{Q^2}=\frac{H}{2} \]
Multiplying both sides by \(Q\):
\[ \frac{DS}{Q}=\frac{QH}{2} \]
The left side is annual ordering cost:
\[ =\frac{DS}{Q} \]
The right side is annual holding cost:
\[ =\frac{QH}{2} \]
Therefore:
\[ \boxed{\text{Annual Ordering Cost}=\text{Annual Holding Cost}} \]
at the basic EOQ.
This equality is a result of the optimization, not an independent assumption of the model.
Important Note About Total Cost
If the unit purchase price is constant, annual purchase cost is:
\[ DC \]
where \(C\) is the unit purchase price.
The complete annual inventory cost can therefore be written as:
\[ TC_{\text{total}}=DC+\frac{D}{Q}S+\frac{Q}{2}H \]
Because \(DC\) is constant with respect to \(Q\), it does not affect the value of EOQ.
Therefore, the optimization can be performed using the relevant cost function:
\[ TC(Q)=\frac{D}{Q}S+\frac{Q}{2}H \]
This distinction is important in Operations Research because EOQ minimizes the relevant cost components that change with order quantity.
EOQ Revision Table
| Concept | Formula / Meaning | Exam Point |
|---|---|---|
| Annual Demand | \(D\) | Total units required per year |
| Order Quantity | \(Q\) | Units ordered each time |
| Ordering Cost | \(S\) | Cost incurred per order |
| Holding Cost | \(H\) | Annual cost of holding one unit |
| Number of Orders | \(\frac{D}{Q}\) | Orders per year |
| Average Inventory | \(\frac{Q}{2}\) | Basic EOQ model |
| Annual Ordering Cost | \(\frac{D}{Q}S\) | Decreases as \(Q\) increases |
| Annual Holding Cost | \(\frac{Q}{2}H\) | Increases as \(Q\) increases |
| Total Relevant Cost | \(\frac{D}{Q}S+\frac{Q}{2}H\) | Cost minimized by EOQ |
| EOQ | \(\sqrt{\frac{2DS}{H}}\) | Economical order quantity |
| Cost at EOQ | Ordering Cost = Holding Cost | Important result |
| EOQ Decision | How much to order | Not when to order |
| Reorder Point | \(ROP=dL\) in a simple deterministic case | Helps determine when to order |
| Basic Model | Known/stable demand, constant costs, no planned shortages | Know the assumptions |
EOQ: Formula Relationships to Remember
The basic EOQ formula is:
\[ EOQ=\sqrt{\frac{2DS}{H}} \]
Therefore:
- If \(D\) increases, EOQ increases.
- If \(S\) increases, EOQ increases.
- If \(H\) increases, EOQ decreases.
These relationships are useful for conceptual questions as well as numerical problems.
Practice Problem for Students
§ Example 1:
A company uses 12,000 units of a component per year.
The ordering cost is 80 currency units per order, and the annual holding cost is 5 currency units per unit.
Find:
- Economic Order Quantity
- Number of orders per year
- Annual ordering cost at EOQ
- Annual holding cost at EOQ
Solution
Given:
\[ D=12,000 \]
\[ S=80 \]
\[ H=5 \]
1. Calculate EOQ
Use:
\[ EOQ=\sqrt{\frac{2DS}{H}} \]
Substitute the values:
\[ EOQ=\sqrt{\frac{2(12,000)(80)}{5}} \]
\[ EOQ=\sqrt{384,000} \]
\[ EOQ\approx619.68 \]
Therefore:
\[ \boxed{EOQ\approx620\text{ units}} \]
2. Calculate the Number of Orders per Year
\[ N=\frac{D}{EOQ} \]
Using the calculated EOQ:
\[ N=\frac{12,000}{619.68} \]
\[ N\approx19.37 \]
Therefore, the model gives approximately:
\[ \boxed{19.37\text{ orders per year}} \]
In practice, the actual ordering schedule may need to be adjusted to whole orders and operational constraints.
Important Note
“After calculating EOQ, we use \(Q=EOQ\) to calculate the number of orders and the annual ordering and holding costs at the optimal order quantity.”
3. Calculate Annual Ordering Cost at EOQ
\[ \text{Annual Ordering Cost}=\frac{D}{Q}S \]
Using \(Q=EOQ=619.68\):
\[=\frac{12,000}{619.68}\times80 \]
\[ \approx1,549.19 \]
Therefore:
\[ \boxed{\text{Annual Ordering Cost}\approx1,549.19} \]
4. Calculate Annual Holding Cost at EOQ
\[ \text{Annual Holding Cost}=\frac{Q}{2}H \]
\[=\frac{619.68}{2}\times5 \]
\[ \approx1,549.19 \]
Therefore:
\[ \boxed{\text{Annual Holding Cost}\approx1,549.19} \]
As expected:
\[ \boxed{\text{Annual Ordering Cost}=\text{Annual Holding Cost}} \]
This confirms the basic EOQ result.
§ Example 2
A company has an annual demand of 20,000 units. The ordering cost is 100 currency units per order, and the annual holding cost is 8 currency units per unit.
Find:
- EOQ
- Number of orders per year
- Annual ordering cost at EOQ
- Annual holding cost at EOQ
Answer
Given:
\[ \text{Annual Demand}=D=20,000 \]
\[ \text{Ordering Cost}=S=100 \]
\[ \text{Annual holding cost}=H=8 \]
1. Calculate EOQ
Use:
\[ EOQ=\sqrt{\frac{2DS}{H}} \]
Substitute the values:
\[ EOQ=\sqrt{\frac{2(20,000)(100)}{8}} \]
\[ EOQ=\sqrt{500,000} \]
\[ \boxed{EOQ\approx707.11\text{ units}} \]
Therefore, the economical order quantity is approximately:
\[ \boxed{707\text{ units per order}} \]
2. Calculate Number of orders per year
Number of orders:
\[ N=\frac{D}{EOQ} \]
Using the calculated EOQ:
\[ N=\frac{20,000}{707.11} \approx28.28 \]
So:
\[ \boxed{28.28\text{ orders per year}} \]
Important Note
“After calculating EOQ, we use \(Q=EOQ\) to calculate the number of orders and the annual ordering and holding costs at the optimal order quantity.”
3. Calculate Annual Ordering Cost at EOQ
\[ \text{Annual Ordering Cost}=\frac{D}{Q}S \]
Using \(Q=EOQ=707.11\), so annual ordering cost:
\[ =\frac{20,000}{707.11}\times100 \approx2,828.43 \]
4. Calculate Annual holding cost at EOQ
\[ \text{Annual Holding Cost}=\frac{Q}{2}H \]\[ =\frac{707.11}{2}\times8 \approx2,828.43 \]
Therefore:
\[\boxed{\text{Annual Ordering Cost}=\text{Annual Holding Cost} \approx2,828.43} \]
This confirms the basic EOQ result.
Quick Exam Revision
Before an EOQ numerical, identify these three values:
\[ D=\text{Annual Demand} \]
\[ S=\text{Ordering Cost per Order} \]
\[ H=\text{Annual Holding Cost per Unit} \]
Then use:
\[ \boxed{EOQ=\sqrt{\frac{2DS}{H}}} \]
For a complete answer, remember:
\[ \boxed{\text{Number of Orders}=\frac{D}{EOQ}} \]
\[ \boxed{\text{Average Inventory}=\frac{EOQ}{2}} \]
\[ \boxed{\text{Annual Ordering Cost}=\frac{D}{EOQ}S} \]
\[ \boxed{\text{Annual Holding Cost}=\frac{EOQ}{2}H} \]
At the basic EOQ:
\[ \boxed{\text{Annual Ordering Cost}=\text{Annual Holding Cost}} \]
The most important conceptual distinction is:
EOQ tells us how much to order; Reorder Point helps determine when to order.
An Important Result at EOQ
At the optimal EOQ in the basic model:
\[ \text{Annual Ordering Cost}=\text{Annual Holding Cost} \]
That means:
\[ \frac{D}{Q}S=\frac{Q}{2}H \]
This equality is a useful way to understand why the total relevant cost is minimized at EOQ.
However, remember an important point:
EOQ is an order quantity, not a reorder point.
EOQ answers:
How much should we order?
The Reorder Point (ROP) answers:
When should we place the order?
These are two different inventory decisions.
Numerical Example of EOQ
Suppose a company has the following inventory requirements:
- Annual demand \(D=10,000\) units
- Ordering cost \(S=200\) per order
- Holding cost \(H=5\) per unit per year
Find the EOQ.
Step 1: Write the Formula
\[ EOQ=\sqrt{\frac{2DS}{H}} \]
Step 2: Substitute the Values
\[ EOQ=\sqrt{\frac{2(10,000)(200)}{5}} \]
Step 3: Calculate
\[ EOQ=\sqrt{800,000} \]
\[ EOQ\approx894.43 \]
Therefore:
\[ \boxed{EOQ\approx894\text{ units}} \]
So, under the assumptions of this example, the economically optimal order size is approximately 894 units per order.
Checking the EOQ Result
A useful exam technique is to check whether the result makes sense.
Number of Orders per Year
\[ \frac{D}{EOQ}=\frac{10,000}{894.43} \approx11.18 \]
So the company would place approximately 11.18 orders per year under the model.
Annual Ordering Cost
\[ \frac{10,000}{894.43}\times200 \approx2,236.07 \]
Annual Holding Cost
\[ \frac{894.43}{2}\times5 \approx2,236.07 \]
The two costs are equal, as expected at the basic EOQ.
This is also a useful sanity check when solving EOQ numerical problems.
A Real-Life Example: A Stationery Shop
Consider a stationery shop that regularly sells notebooks.
Suppose the shop expects a fairly stable annual demand for a particular notebook.
If the shop orders only a small quantity each time, it will have to place many orders during the year.
Small order quantity
→ more orders
→ more ordering activity
→ higher annual ordering cost
Now suppose the shop purchases a very large quantity at once.
Large order quantity
→ fewer orders
→ lower annual ordering cost
→ more inventory held in the shop
→ higher storage and capital-related holding costs
EOQ helps the shop find a quantity between these two extremes.
The important lesson is:
EOQ is not simply about buying in bulk. It is about finding an economically appropriate order size by balancing ordering and holding costs.
This is why EOQ is useful as a basic inventory decision model.
What Happens When the Order Quantity Is Too Small or Too Large?
| Order Quantity | Ordering Cost | Holding Cost | Likely Effect |
|---|---|---|---|
| Very small | High | Low | Total cost may be high |
| Near EOQ | Balanced | Balanced | Total relevant cost is minimized |
| Very large | Low | High | Total cost may be high |
This table gives the basic economic logic behind EOQ.
How Changes in the Inputs Affect EOQ
The EOQ formula is:
\[ EOQ=\sqrt{\frac{2DS}{H}} \]
From this formula:
If Annual Demand \(D\) Increases
EOQ increases.
A business with greater annual demand generally needs a larger economic order quantity, assuming the other parameters remain unchanged.
If Ordering Cost \(S\) Increases
EOQ increases.
When placing an order becomes more expensive, it becomes economically attractive to place fewer orders with a larger quantity in each order.
If Holding Cost \(H\) Increases
EOQ decreases.
When holding inventory becomes more expensive, smaller order quantities become more attractive.
This provides an important managerial insight:
EOQ is not just a formula to calculate a number. It shows how inventory decisions respond to changes in cost and demand.
Assumptions of the Basic EOQ Model
The classical EOQ model is based on simplifying assumptions. These assumptions are important because the formula should not be applied blindly to every inventory situation.
The basic model generally assumes:
1. Demand is Known and Relatively Constant
Annual demand \(D\) is known or can be estimated with sufficient certainty.
2. Lead Time is Known and Constant
The time between placing an order and receiving it is assumed to be known and stable.
3. Replenishment is Immediate for the Basic Model
The standard basic EOQ model assumes that the entire order becomes available at once.
4. Ordering Cost is Constant
The cost of placing one order is assumed to remain constant.
5. Holding Cost is Constant
The annual holding cost per unit is assumed to remain constant.
6. Unit Purchase Price is Constant
The basic model does not consider quantity discounts or changes in unit purchase price.
7. Shortages are Not Allowed
The basic EOQ model assumes that inventory is replenished before a shortage occurs.
8. A Single Item is Considered
The simplest EOQ model considers one inventory item independently.
9. Demand and Replenishment Conditions are Sufficiently Stable
The model is designed for a relatively predictable inventory environment rather than highly uncertain demand or supply conditions.
These assumptions make the model simple and useful for understanding the basic inventory trade-off.
Ordering Cost in EOQ
Ordering cost is the cost associated with placing and processing an order.
It may include:
- Preparing and processing purchase orders
- Administrative and documentation costs
- Communication costs
- Receiving and inspection activities
- Other fixed costs associated with placing an order
The important point is that the ordering cost in the basic EOQ model is treated as a cost per order, not as a cost per unit purchased.
If:
\[ S=\text{ordering cost per order} \]
and the business places:
\[ \frac{D}{Q} \]
orders per year, then:
\[ \text{Annual Ordering Cost}=\frac{D}{Q}S \]
Holding or Carrying Cost in EOQ
Holding cost is the cost of keeping inventory available for use or sale.
It may include:
- capital or financing cost associated with inventory
- warehouse or storage cost
- insurance
- deterioration and spoilage
- inventory handling
- obsolescence where relevant
If:
\[ H=\text{annual holding cost per unit} \]
and average inventory is:
\[ \frac{Q}{2} \]
then:
\[ \text{Annual Holding Cost}=\frac{Q}{2}H \]
What About Purchase Cost?
A common source of confusion is the purchase cost of the inventory.
If the unit purchase price is constant and there is no quantity discount, the annual purchase cost is:
\[ DC \]
where \(C\) is the unit purchase price.
Since \(DC\) does not change when \(Q\) changes, it does not affect the basic EOQ calculation.
Therefore, the EOQ decision focuses on the relevant ordering and holding costs:
\[ TC(Q)=\frac{D}{Q}S+\frac{Q}{2}H \]
If purchase prices change with order quantity, such as through quantity discounts, a different analysis is required.
EOQ and Reorder Point Are Not the Same
This is one of the most important points for students.
▪️EOQ
EOQ tells us how much to order.
\[ EOQ=\sqrt{\frac{2DS}{H}} \]
▪️Reorder Point
Reorder Point helps determine when an order should be placed.
Under a simple deterministic setting without safety stock:
\[ ROP=dL \]
where:
- \(d\) = demand per unit of time
- \(L\) = lead time in the same unit of time
The actual reorder-point model may need to consider safety stock when demand or lead time is uncertain.
Therefore:
EOQ and ROP solve different inventory questions and should not be treated as interchangeable concepts.
Common Mistakes in EOQ Numerical Problems
EOQ common mistakes for students — Students often know the formula but lose marks because of incorrect data interpretation.
Mistake 1: Mixing Time Units
If demand is given per year, holding cost should normally be expressed per unit per year.
Do not combine annual demand with monthly holding cost without converting the units.
Mistake 2: Confusing Ordering Cost with Annual Ordering Cost
The EOQ formula uses:
ordering cost per order
not total annual ordering cost.
Mistake 3: Forgetting the Unit of the Answer
EOQ is a quantity, so the final answer should be written in units.
Mistake 4: Treating EOQ as Reorder Point
EOQ tells us how much to order, not by itself when to order.
Mistake 5: Adding Shortage Cost to the Basic EOQ Formula
The basic EOQ model does not include planned shortages.
A shortage or backorder model requires a different formulation.
Mistake 6: Ignoring Quantity Discounts
If the problem gives different purchase prices for different order quantities, the basic EOQ formula alone may not be sufficient.
Advantages of EOQ
The basic EOQ model has several useful features.
1. It Provides a Quantitative Order-Size Decision
Instead of choosing an order quantity arbitrarily, the business can use a mathematical model.
2. It Balances Ordering and Holding Costs
EOQ captures the basic trade-off between ordering frequently and holding large quantities.
3. It Is Simple to Calculate
The basic EOQ formula is straightforward and can be used easily when its assumptions are reasonably appropriate.
4. It Provides a Useful Starting Point for Inventory Planning
Even when a real inventory system is more complex, the basic EOQ model provides a useful benchmark.
5. It Helps Students Understand Inventory-Cost Trade-Offs
EOQ is an important introductory model because it connects mathematical optimization with a practical management decision.
Limitations of EOQ
The basic EOQ model is useful, but it is not a complete description of every real-world inventory system.
1. Demand May Not Be Constant
Real demand can vary because of seasonality, market conditions, promotions, or unexpected changes.
2. Ordering Costs May Not Remain Constant
Actual procurement costs may vary between suppliers, orders, or circumstances.
3. Holding Costs May Be Difficult to Estimate
The true cost of storage, capital, insurance, deterioration, and obsolescence may change over time.
4. Basic EOQ Does Not Allow Planned Shortages
If shortages or backorders are intentionally permitted, a different inventory model is required.
5. Quantity Discounts Are Not Included
If suppliers offer lower prices for larger orders, the basic EOQ calculation must be modified or compared with the relevant price-break quantities.
6. Supply Conditions May Be Uncertain
Supplier delays, transportation problems, or variable lead times may make the basic assumptions unrealistic.
7. Multiple-Item Inventory Creates Additional Constraints
A business may have limited warehouse space, limited working capital, or many products competing for the same resources.
Therefore:
EOQ should be viewed as a basic analytical model rather than a universal rule for every inventory decision.
When Should the Basic EOQ Model Be Used?
The basic EOQ model is most appropriate when:
- demand is reasonably stable and predictable
- ordering cost can be estimated
- holding cost can be estimated
- the unit purchase price is constant
- shortages are not planned
- replenishment conditions are reasonably stable
- the item can be analyzed independently
When these conditions do not hold, an extended or different inventory model may be more appropriate.
EOQ in Real Business Decisions
EOQ is especially useful for items that have relatively predictable demand and meaningful ordering and holding costs.
For example, consider a manufacturer that regularly uses a standard component in production.
If the manufacturer orders too frequently:
- procurement activity increases
- administrative work increases
- ordering costs increase
If it orders very large quantities:
- more capital is tied up in inventory
- storage requirements increase
- the risk of deterioration or obsolescence may increase
EOQ provides a benchmark order quantity that balances these two types of costs under the model's assumptions.
In actual inventory management, however, managers may also need to consider:
- safety stock
- supplier reliability
- lead-time variability
- warehouse capacity
- quantity discounts
- service-level requirements
- cash-flow constraints
- demand forecasts
This is why EOQ is best understood as a decision model, not as a rigid rule.
Basic EOQ vs. Other Inventory Models
The basic EOQ model is only one member of a larger family of inventory models.
| Situation | Possible Model or Approach |
|---|---|
| Known and relatively constant demand | Basic EOQ |
| Planned shortages or backorders | EOQ with shortages |
| Quantity discounts | Quantity-discount inventory model |
| Internal production instead of instant replenishment | Economic Production Quantity (EPQ) |
| Uncertain demand | Probabilistic inventory models |
| Variable lead time | Models incorporating lead-time uncertainty |
| Multiple inventory items and constraints | Multi-item inventory models |
This distinction is important because a formula should be selected according to the characteristics of the inventory problem.
EOQ: A Quick Conceptual Summary
Remember these five points:
- EOQ tells us how much to order.
- Ordering cost decreases as order quantity increases.
- Holding cost increases as order quantity increases.
- EOQ minimizes the combined relevant ordering and holding costs in the basic model.
- EOQ does not automatically tell us when to place the order.
EOQ Formula for Quick Revision
\[ \boxed{EOQ=\sqrt{\frac{2DS}{H}}} \]
Where:
\[ D=\text{Annual demand} \]
\[ S=\text{Ordering cost per order} \]
\[ H=\text{Annual holding cost per unit} \]
Annual ordering cost:
\[ \boxed{\frac{D}{Q}S} \]
Average inventory:
\[ \boxed{\frac{Q}{2}} \]
Annual holding cost:
\[ \boxed{\frac{Q}{2}H} \]
Total relevant annual cost:
\[ \boxed{TC(Q)=\frac{D}{Q}S+\frac{Q}{2}H} \]
At the basic EOQ:
\[ \boxed{\text{Annual Ordering Cost}=\text{Annual Holding Cost}} \]
EOQ: Exam-Friendly Answer
Definition
Economic Order Quantity (EOQ) is the order quantity that minimizes the relevant total cost of ordering and holding inventory under the assumptions of the basic EOQ model.
Formula
\[ \boxed{EOQ=\sqrt{\frac{2DS}{H}}} \]
Main Assumptions
- Demand is known and relatively constant.
- Ordering cost is constant.
- Holding cost is constant.
- Unit purchase price is constant.
- Lead time is known and stable.
- Replenishment is immediate in the basic model.
- Shortages are not allowed.
- Quantity discounts are not considered.
Main Objective
To determine the economical order quantity that minimizes the relevant ordering and holding costs.
Frequently Asked Questions About EOQ
What does EOQ stand for?
EOQ stands for Economic Order Quantity.
What does EOQ determine?
EOQ determines the economical quantity to order at one time under the assumptions of the basic EOQ model.
What is the EOQ formula?
\[ EOQ=\sqrt{\frac{2DS}{H}} \]
where \(D\) is annual demand, \(S\) is ordering cost per order, and \(H\) is annual holding cost per unit.
What is Optimal Order Quantity?
Optimal Order Quantity is:
\[ Q^* = EOQ \]
Or, even more clearly:
\[ \boxed{Q^* = EOQ = \sqrt{\frac{2DS}{H}}} \]
Here, \(Q^*\) means the optimal order quantity, and this optimal value is called EOQ (Economic Order Quantity).
For your article, I recommend writing:
At the optimal order quantity, \(Q = Q^*\), and \(Q^*\) is called the EOQ.
This is mathematically precise and avoids saying that every \(Q\) is EOQ.
Note:
After calculating EOQ, we use \(Q=Q^*=EOQ\) to calculate the number of orders and the annual ordering and holding costs at the optimal order quantity.
Why does EOQ balance ordering and holding costs?
Ordering more frequently increases ordering cost, while ordering larger quantities increases holding cost. EOQ finds the order quantity at which the combined relevant cost is minimized.
Are ordering cost and holding cost equal at EOQ?
Yes. In the basic EOQ model, annual ordering cost equals annual holding cost at the optimal EOQ.
Does EOQ tell us when to reorder?
No. EOQ tells us how much to order. Reorder point determines when to place the order.
Does the basic EOQ model allow shortages?
No. The basic EOQ model assumes that shortages are not allowed. Models with planned shortages or backorders use additional assumptions and a modified formulation.
Does EOQ consider quantity discounts?
The basic EOQ model assumes a constant unit purchase price and therefore does not directly account for quantity discounts. A separate quantity-discount analysis is required.
Is EOQ useful in real life?
Yes, when its assumptions are reasonably appropriate. In more complex situations, managers may need to combine EOQ with reorder-point calculations, safety stock, demand forecasting, supplier considerations, or other inventory models.
Final Takeaway
Economic Order Quantity (EOQ) is a fundamental inventory model in Operations Research. It provides a simple mathematical way to balance ordering and holding costs and determine an economical order size.
The most important idea is not simply to memorize:
\[ EOQ=\sqrt{\frac{2DS}{H}} \]
Instead, understand the logic behind it:
Small orders mean more frequent ordering. Large orders mean more inventory to hold. EOQ finds the economical balance between these two costs under the assumptions of the basic model.
For examination purposes, remember:
EOQ = How much to order
ROP = When to order
Basic EOQ = Stable or known demand + ordering cost + holding cost + no planned shortages + constant unit price
Once this basic model is understood, more advanced inventory models such as EOQ with shortages, quantity-discount models, Economic Production Quantity (EPQ), and probabilistic inventory models can be studied more easily.
References
- Hamdy A. Taha, Operations Research: An Introduction, 10th Edition, Pearson.
- Frederick S. Hillier and Gerald J. Lieberman, Introduction to Operations Research, McGraw-Hill Education.
- J. K. Sharma, Operations Research: Theory and Applications, Macmillan India.
- Kanti Swarup, P. K. Gupta and Man Mohan, Operations Research, Sultan Chand & Sons.
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