---
title: "Derivation of economic order quantity"
subject: "Industrial Engineering and Management"
chapter: 4
key: "semester_6/industrial_engineering_and_management:eoq-model-derivation-and-assumptions"
source: https://www.bctengineer.com/q/semester_6/industrial_engineering_and_management/eoq-model-derivation-and-assumptions
site: BCT Engineering
---

# Derivation of economic order quantity

Industrial Engineering and Management · Chapter 4: Inventory Control

- Asked in: year not given, 6 marks
- Video: [Derivation of Economic Order Quantity(EOQ) | Must Know | Accounts | Production | Finance](https://www.youtube.com/watch?v=gFbvAivvU3g) — Simplify Concepts
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/industrial_engineering_and_management/eoq-model-derivation-and-assumptions
- Whole chapter: https://www.bctengineer.com/md/semester_6/industrial_engineering_and_management/chapter-4

## Question

Derive the formula for economic order quantity (EOQ) for the basic model, stating the assumptions. Also show that at EOQ the ordering cost equals the holding cost.

## Short answer

EOQ = sqrt(2DS/H) derived by minimizing total cost (ordering + holding) assuming constant demand, lead time, costs, and instantaneous receipt. At EOQ, annual ordering cost equals annual holding cost, each = sqrt(DSH/2).

## Answer

**Economic order quantity** is the order size that minimises the total annual inventory cost (ordering plus holding).

#### Assumptions
- Demand $D$ per year is known and constant.
- Lead time is constant and no shortage is allowed.
- Whole order of $Q$ units arrives at once.
- Ordering cost $S$ per order and holding cost $H$ per unit per year are constant.
- Price per unit does not depend on the quantity (no discount).

#### Derivation
Inventory falls uniformly from $Q$ to 0, so the average inventory is $Q/2$.

$$
\text{Annual ordering cost} = \frac{D}{Q} S, \qquad \text{Annual holding cost} = \frac{Q}{2} H
$$

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

(The purchase cost $DC$ is constant and is left out of the minimisation.) For minimum cost, differentiate with respect to $Q$:

$$
\frac{dTC}{dQ} = -\frac{DS}{Q^2} + \frac{H}{2} = 0 \;\Rightarrow\; Q^2 = \frac{2DS}{H}
$$

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

The second derivative $2DS/Q^3$ is positive, so this is a minimum.

#### Ordering cost equals holding cost at EOQ
From $Q^{*2} = 2DS/H$, we get $\dfrac{D}{Q^*}S = \dfrac{Q^*}{2}H$ because both equal $\sqrt{DSH/2}$. Hence

$$
TC_{min} = 2\sqrt{\frac{DSH}{2}} = \sqrt{2DSH}
$$

#### Related results
Number of orders per year $N = D/Q^*$; time between orders $T = Q^*/D$; reorder point $= d \times L$ (daily demand times lead time). The total cost curve is flat near the EOQ, so small changes in $Q$ change cost very little.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which is an assumption of the basic EOQ model?
   A. Shortages allowed
   B. Quantity discounts
   C. Demand known and constant, whole order arrives at once
   D. Variable lead time
   Answer: C. Demand and costs are constant, no shortage, no discount. (Key point: Assumptions)

2. What is the total annual cost expression?
   A. TC = (D/Q)S + (Q/2)H
   B. TC = DS + QH
   C. TC = (Q/D)S + (D/2)H
   D. TC = (D/Q)H + (Q/2)S
   Answer: A. Ordering cost is (D/Q)S and holding cost is (Q/2)H, since average inventory is Q/2. (Key point: Total cost)

3. What is the EOQ formula?
   A. sqrt(DSH/2)
   B. sqrt(2DH/S)
   C. sqrt(2DS/H)
   D. 2DS/H
   Answer: C. Setting dTC/dQ = 0 gives Q^2 = 2DS/H. (Key point: EOQ)

4. What is true of ordering and holding costs at EOQ?
   A. Ordering is double holding
   B. They are equal
   C. Holding is zero
   D. Ordering is zero
   Answer: B. Both equal sqrt(DSH/2), so TC_min = sqrt(2DSH). (Key point: Equal costs)

## Related in Industrial Engineering and Management

- [EOQ, total cost and reorder point numerical](https://www.bctengineer.com/question.md?key=semester_6%2Findustrial_engineering_and_management%3Aeoq-numerical-reorder-point) — Industrial Engineering and Management, chapter 4
- [MRP explosion with lot-for-lot ordering](https://www.bctengineer.com/question.md?key=semester_6%2Findustrial_engineering_and_management%3Amrp-net-requirement-calculation) — Industrial Engineering and Management, chapter 5

## Related in other subjects

- [Define economic order quantity](https://www.bctengineer.com/question.md?key=semester_8%2Fconstruction_management%3Adefine-eoq) — Construction Management, chapter 3
- [Material classification, EOQ, GiT and re-order level](https://www.bctengineer.com/question.md?key=semester_8%2Fconstruction_management%3Aclassification-eoq-gitt-2079) — Construction Management, chapter 3
- [EOQ and reorder point for rebar](https://www.bctengineer.com/question.md?key=semester_8%2Fconstruction_management%3Aeoq-rebar-2071) — Construction Management, chapter 3
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