SYNOPSIS ON EOQ (For Students’ Seminar)

Economic Order QuantityPrepared for: KIAMS, HariharPrepared by: E.Anand, Lecturer, BIET-MBA Programme

19 October 2009

E.AnandFaculty, BIET-MBA Programme

S S LayoutDavangere, Karanataka, 577004

tel 08192220501

9742318614

elluranand@yahoo.com

To understand Economic Order Quantity [EOQ], it is necessary to understand basic concepts of

inventory control which forms the basis for EOQ. For finance department, inventory means rupee value

that affects balance sheet, profit and loss account but term ‘inventory’ for manufacturing means stock

of raw material, work-in-process, supplies, component parts and finished goods measured in terms of

quantity (kgs, units, metres etc.). It can be measured in rupee value as well as in quantity (units). In

order to deliver the goods to satisfy the customer needs at faster rate, firms usually tend to stock up

the goods.

There are quite a few reasons why firms tend to maintain inventory, as explained by Richard Chase

i) Due to variation in the raw material delivery: This means that the raw material which has to be

supplied by a supplier who is external to a company, some times, cannot maintain the constant lead

time due to various reasons and not only that, there can be other issues like strikes which are in no

way related to both the parties. These situations forces both the manufacturer & distributor to

maintain inventories. This reason to hold inventory is with respect to raw material.

ii) Independence of operations: This reason to hold inventory is with respect to Work-in-process

(WIP). At each work centre, to achieve independence from upstream activities in case of breakdown,

inventory in the form of buffer is maintained.

iii) Variation in product demand: This reason to hold inventory is with respect to finished goods. The

demand of the product is totally independent which makes it highly uncertain by nature. To counter

this uncertainty firms tend to stock up goods in quite scientific manner - where enough goods are

stocked up to check any customer’s satisfaction being unmet with while at the same time it should

minimize the cost incurred by the company.

iv) Economics of purchase order size: This inventory is the result of firms’ intentions to take

advantage of economies of scale and discounts in bulk purchases.

i) Having too much inventory of some product

ii) But experience stock-outs of other products thus having dissatisfied customers

iii) Do not know what is in stock

iv) Cannot trace the item required at the time it is needed the most

Page 1

Fundamentals of Inventory

Problems Firms’ Encounter with Inventory

The first two points refer to inventory management aspects while the other two refer to inventory

control aspects. Inventory management comprises inventory planning, order quantity while inventory

control is controlling the inventory once it is in-house or in warehouse like receipt of goods,

arrangement of goods, issuing to concern departments and maintaining & updating the records. The

main problem firms face is ‘where to start?’. It is always better to start looking what is in-house i.e.

Inventory control and then moving towards aspects concerning inventory management. Inventory

control is taken care, usually, by the computer-based inventory management systems.

As already pointed out in the above paragraph, inventory management usually includes inventory

planning aspects. Inventory planners or firms are mostly interested in knowing answers to basic

questions “When to order?” and “How much to order?”. The answer for the first question helps to fix

the Re-order point or level and the answer for the second question helps to fix the “Economic Order

Quantity - EOQ”. There are basically two types of inventory systems:

i) Single Period Inventory System: This is the inventory system which gains significance in situations

where the inventory is of highly perishable type. “Today’s inventory cannot be used tomorrow” type

situations. Famous example of such situation is ‘Paperboy Problem’ where todays newspaper for the

paperboy will be useless tomorrow and to decide on how much inventory to be maintained poses a

big challenge.

ii) Multiperiod Inventory System: This inventory system gains significance in most of the business

situations as most of the firms fall in this category where inventory can carried over certain periods.

This system can be further classified into two types:

✴‘Q’ Inventory System or Fixed-Order Quantity System

✴‘P’ Inventory System or Fixed-Time Period System

✴

As the name suggests, in this system the order quantity is fixed for all the orders. The time to place an

order is decided by pre-fixed or calculated re-order level. It is typically like this “If the inventory level falls

below or reaches 20 units, place an order for 250 nos.”. This order quantity which is fixed is the

Page 2

Inventory Management

Fixed-Order Quantity System or ‘Q’ - System

optimum order quantity or EOQ - Economic Order Quantity. It is optimum or economical because at

this quantity which is calculated in a scientific manner checks that order size is enough to meed the

customer needs and at the same time it minimizes the inventory carrying cost. It is necessary to

understand two types of inventory costs and the assumptions to know how EOQ is arrived at.

Different types of inventory costs are:

i) Carrying costs: This includes interest on the loan taken for inventory, insurance of the stock,

obsolescence, theft, pilferage and the most important storage costs which should be carefully

handled to include the right costs and to exclude those costs which do not directly affect the

inventory in-hand.

ii) Ordering costs: This includes cost of placing an order, approval steps, cost to process the receipt,

cost of inspection, invoice processing, in some cases a part of freight charges, expediting costs.

iii) Set-up costs: This includes time to initiate the order, time to pick and issue the necessary

components, all production scheduling time, machine set-up time, scrap and tooling charges also to

be included in this costs.

Certain assumptions are made to calculate EOQ under certainty, though rarely realistic but gives a

good starting point for uncertain conditions. Assumptions are:

i) Demand for the product is constant

ii) Lead time is constant

iii) Price per unit of product is constant

iv) Inventory carrying cost is based on average inventory

v) Ordering costs are constant

vi) No backorders (all the demand is instantly satisfied)

Page 3

Inventory Costs

Assumptions

The profit motive of a firm, with respect to holding inventory, is achieved by reducing the costs

associated with inventory. The behavior of carrying or holding costs and the ordering or set-up cost

against the quantity ordered is shown in the graph in the previous page. It is obvious that while the

holding cost increases with quantity ordered, the ordering costs decreases with the bigger order size.

At the same time it can also be seen that purchase order cost is constant as it is not a function of the

quantity but of cost alone i.e. Purchase cost = D.C where D = annual demand and C = cost per unit of

inventory. Now look at the total cost curve that connects the points obtained by adding the

simultaneous three types of costs (or in fact only two costs as purchase cost doesn’t affect the total

cost curve as it is constant throughout). The slope of the curve at the trough where the blue line

starts downwards is zero. This extending blue line touching the x - axis can be seen cutting through the

intersection point of two costs - Carrying cost curve & Ordering cost curve. This is the point at which

they both are equal in the sense the point on x-axis gives us the order quantity that minimizes both

types of costs. This optimal quantity is called as Economic Order Quantity (EOQ). Now it is easy to

understand how to arrive at EOQ formula to make easy for the firms to plan their inventory properly.

Page 4

TC Total Cost

(D/Q) SOrdering Cost

Cos

t

Order Quantity Size

(Q/2) H Holding Cost

DCPurchase Cost

Qopt or EOQ

Inventory Costs Trade-off

From the graph, we know that (as already explained)

Total cost TC = Purchase cost + Carrying cost + Ordering cost

Purchase cost = D.C where D = annual demand in units, C = Cost per unit

Carrying cost = [Q/2] * H where Q = Optimum Quantity or EOQ; H = Holding/carrying cost per unit or

calculated as percentage of unit cost i.e. H = iC where i= percent carrying cost; Q/2 = average

inventory carried.

Ordering cost = [D/Q] * S where D = annual demand in units; Q = EOQ and S = order cost or set-up

cost

Substituting the values, we get,

TC = D.C + [Q/2] * H + [D/Q] * S

On the total cost curve, the minimal point is where the slope of curve is zero. Taking derivative of the

total cost function with respect to Q and setting this equal to zero, we get,

dTC = d DC + d [Q/2]*H + d [D/Q] * S = 0

dQ dQ dQ dQ

0 = 0 + H/2 + (-DS/Q2)

Qopt = EOQ = √2DS/H

Reorder Point ‘R’ = d L, where d = average daily demand (constant) and L = lead time in days

(constant)

One of the assumptions was that demand is constant and we know that it is not so in reality. Suppose,

if we consider the fact that demand is not constant but varying then how would the above EOQ and

reorder level change?. To accomodate this variability in demand, many firms maintain a safety stock.

Page 5

Safety Stock

Economic Order Quantity (EOQ)

This safety stock provides security against any stockouts. How much should be the safety stock level,

how it is fixed by the firms?. Many firms follow different ways of maintaining the stock. Few of them are:

i) It can be supply of few number of weeks

ii) It can be ½ lead time usage method

iii) It can be just trial and error method trying to fix the best safety stock level

The above mentioned methods are all vague methods to arrive at a stock level. The recommended way

of deciding the quantity to be maintained as safety stock is by Probability Approach.

Probability Approach: Suppose the expected demand for next month is 150 units with a standard

deviation of 30 units. Then if a firm wants to be 85% confident that it would not run out of stock, then

the safety stock it should maintain is given by,

To be 85% confident, we have to find ‘x’ value through ‘z’ value. We know that

z = (x - μ) / σ

From the normal probability distribution table, we get z = 1.04 at 85% confidence interval. Now

substituting the values in the above equations, we get

1.04 = (x - 150) / 30

x = 1.04 * 30 +150

x = 181.2 or 181

So, for a firm to be 85% confident when the average expected demand is 150 units with a standard

deviation of 30 units, the safety stock to be maintained is 181 - 150 = 31 units.

Reorder Point: ‘R’ value is rewritten from

R = d . L to

R = d . L + z.σL

Page 6

0.5 0.35

0.15

μ = 150

σ = 30 x

Though EOQ was formulated under certain conditions, it gives enough room to modify to suit the real

conditions under which inventory is handled. As it is shown above that when the first assumption

(highlighted in orange color) is turned into reality by changing into “Demand is not constant”, how EOQ

model suits itself into the real conditions by considering the probability value. It is quite possible to

eliminate assumptions one by one to match EOQ to reality. What if “Price of the unit is constant”?,

What if “Lead time is not constant”? etc can be found.

There are many other aspects of EOQ which usual text books do not teach. This would make the topic

even more interesting to present.

Page 7

Concluding Remarks