Production Management Assignment

The Economic Order Quantity Model

Name: Stita Pragyna Misra

Class: PGDM III (2011-13)

Roll No. : 156

Welingkar Institute of Management Development and Research

The Economic Order Quantity (EOQ) Model

Overview:

Economic order quantity is the order quantity that minimizes total inventory holding costs and

ordering costs. It is one of the oldest classical production scheduling models. The framework

used to determine this order quantity is also known as Wilson EOQ Model or Wilson Formula.

The model was developed by Ford W. Harris in 1913, but R. H. Wilson, a consultant who

applied it extensively, is given credit for his in-depth analysis.

The economic order-quantity model considers the tradeoff between ordering cost and storage

cost in choosing the quantity to use in replenishing item inventories. A larger order-quantity

reduces ordering frequency, and, hence ordering cost/month, but requires holding a larger

average inventory, which increases storage (holding) cost/month. On the other hand, a smaller

order-quantity reduces average inventory but requires more frequent ordering and higher

ordering cost/month. The cost- minimizing order-quantity is called the Economic Order Quantity

(EOQ).

The EOQ model is the best known and most fundamental inventory decision model. Although

this model is too over simplified to represent most real world situation, it is nevertheless, an

excellent starting point from which to develop complex and more realistic inventory decision

models.

Historical Background

The EOQ model and formula are attributed to Ford Whitman Harris ( Harris, 1913 ), and

originally published in 1913 in Factory, The Magazine of Management. However, despite this

magazine’s wide circulation among manufacturing managers, Harris’ article had been lost until it

was found by Erlenkotter. In the years between Harris’ article being lost and found, the EOQ

model, and, in particular, the square-root equation, (4), was attributed to others, some calling it

the “Wilson lot size formula,” others calling it “Camp’s formula.” Erlenkotter (1989) describes

the early development of the EOQ literature and sketches the life of Harris as an engineer,

inventor, author, and patent attorney.

Where: A = Demand for the year

Cp = Cost to place a single order

Ch = Cost to hold one unit inventory for a year

Assumptions:

As is the case with all models, the validity of EOQ

model depends on a number of assumption are as follows:-

i. Inventory is replenished when the inventory is exactly equal to zero (no shortage).

ii. Demand (usage) rate is known and is constant.

iii. Lead time, the interval between the order is placed and the time it is received, is known

and constant.

iv. Carrying cost is linear throughout the entire inventory and varies with average inventory.

v. Price of the product does not depend on the quantity purchased. (The purchase price of

the item is constant i.e. no discount is available)

vi. Ordering of the products is independent of ordering other products.

vii. The ordering cost is constant.

viii. The replenishment is made instantaneously, the whole batch is delivered at once.

ix. Only one product is involved.

The above graph illustrates the variation of the inventory level over time for the classic EOQ

model. The down ward sloping curve indicates that the inventory level is being reduced at a

constant rate over consumption time. Inventory level equal to Q when each new order is

physically received into store.

Furthermore, the inventory is gradually depleted until it reaches zero just at the point when the

new order is received. The average inventory (Q/2) is, thus, equal to one – half

number of this within the same period.

The economic order quantity will be optimal when all five assumptions are satisfied. In reality,

few situations are so simple. Nonetheless, the EOQ is often a reasonable approxima¬tion of the

appropriate lot size, even when several of the assumptions do not quite apply. Here are some

guidelines on when to use or modify the EOQ.

Don't use the EOQ

- If you use the "make-to-order" strategy and your customer specifies the entire order be

delivered in one shipment

- If the order size is constrained by capacity limitations such as the size of the firm's ovens,

amount of testing equipment, or number of delivery trucks

Modify the EOQ

- If significant quantity discounts are given for ordering larger lots

- If replenishment of the inventory is not instantaneous, which can happen if the items

must be used or sold as soon as they are finished without waiting until the entire lot

has been completed (see Supplement D, "Special Inventory Models," for several useful

modifications to the EOQ)

Use the EOQ

- If you follow a "make-to-stock" strategy and the item has relatively stable demand.

- If your carrying costs and setup or ordering costs are known and relatively stable

THE MODEL FORMULATION

The first step in constructing the inventory model is to define its variable and parameters.

Let:

Q= order quantity

Q* = optimal order quantity

D = annual demand quantity

S = fixed cost per order (not per unit, typically cost of ordering and shipping and handling. This

is not the cost of goods)

H = annual holding cost per unit (also known as carrying cost or storage cost) (warehouse space,

refrigeration, insurance, etc. usually not related to the unit cost)

The Total Cost function

The single-item EOQ formula finds the minimum point of the following cost function:

Total Cost = purchase cost + ordering cost + holding cost

Purchase cost: This is the variable cost of goods: purchase unit price × annual demand quantity.

This is P×D

Ordering cost: This is the cost of placing orders: each order has a fixed cost S, and we need to

order D/Q times per year. This is S × D/Q

Holding cost: the average quantity in stock (between fully replenished and empty) is Q/2, so this

cost is H × Q/2

.

To determine the minimum point of the total cost curve, partially differentiate the total cost with

respect to Q (assume all other variables are constant) and set to 0:

Solving for Q gives Q* (the optimal order quantity):

Therefore:

Q* is independent of P; it is a function of only S, D, H.

Extensions

Several extensions can be made to the EOQ model, including backordering costs and multiple

items. Additionally, the economic order interval can be determined from the EOQ and the

economic production quantity model (which determines the optimal production quantity) can be

determined in a similar fashion.

A version of the model, the Baumol-Tobin model, has also been used to determine the money

demand function, where a person's holdings of money balances can be seen in a way parallel to a

firm's holdings of inventory.

Variations

There are many variations on the basic EOQ model. I have listed the most useful ones below.

· Quantity discount logic can be programmed to work in conjunction with the EOQ formula

to determine optimum order quantities. Most systems will require this additional programming.

· Additional logic can be programmed to determine max quantities for items subject to

spoilage or to prevent obsolescence on items reaching the end of their product life cycle.

· When used in manufacturing to determine lot sizes where production runs are very long

(weeks or months) and finished product is being released to stock and consumed/sold throughout

the production run you may need to take into account the ratio of production to consumption to

more accurately represent the average inventory level.

· Your safety stock calculation may take into account the order cycle time that is driven by

the EOQ. If so, you may need to tie the cost of the change in safety stock levels into the formula.

Example:

A basic problem for businesses and manufacturers is, when ordering supplies, to determine

what quantity of a given item to order. A great deal of literature has dealt with this problem.

Many formulas and algorithms have been created. Of these the simplest formula is the most

used: The EOQ (economic order quantity) or Lot Size formula. The EOQ formula has been

independently discovered many times in the last eighty years. The EOQ formula is simplistic and

uses several unrealistic assumptions. This raises the question, which we will address: given that

it is so unrealistic, why does the formula work so well? Indeed, despite the many more

sophisticated formulas and algorithms available, even large corporations use the EOQ formula.

In general, large corporations that use the EOQ formula do not want the public or competitors to

know they use something so unsophisticated.

It is useful at this point to consider a numerical example. Let’s assume that the demand for

bolts’s is 50 per week. The order cost is Rs.30 (regardless of the size of the order), and the

holding cost is Rs.6 per bolt per week. Plugging these figures into the EOQ formula we get:

This brings up a little mentioned drawback of the EOQ formula. The EOQ formula is not an

integer formula. It would be more appropriate if we ordered bolts by the gallon. Most of the

time, the nearest integer will be the optimal integer amount. In this case, the total inventory cost

Tµ is Rs.134.18 per week when we order 22 bolts. If instead, we order 23 bolts the cost is

Rs.134.22.

A graph of this problem is illuminating: Figure 2. Because the graph is so flat at the optimal

point, there is very little penalty if we order a slightly sub-optimal quantity. We can better

understand the graph if we view the combined graphs of the order costs and the holding costs

given in Figure 3. The basic shapes of all three graphs (total costs, order costs, holding costs) are

always the same. The graph of order costs is a hyperbola; the graph of holding costs is linear;

and as a result the graph of the total costs (Tµ ) is convex. This can also be seen in that the

function increasing and the function

is positive everywhere. If we plug the optimal quantity, Q*, into this last

formula we get:

Ordinarily this last quantity is very small, which indicates that the total cost of inventory Tµ

changes very slowly with Q (in the optimal region). Hence the assumptions of the EOQ model

do not have to be accurate because the problem usually is tolerant of errors. If you study closely

the graphs in Figure 3, it may seem clear to you that their sum, Tµ, reaches a minimum precisely

where the two graphs intersect; that is at the point where order costs and holding costs are equal.

The gives us the equality. Solving that equality is the easiest

way to derive the EOQ formula.

Application:

Cash Management

The EOQ model can be applied to any quantity decision that is repeated over time, where there

is a trade-off between a fixed cost (i.e., EOQ’s order cost K) and a variable cost associated with

that decision (i.e., EOQ’s inventory-holding cost), and where the choice is based on minimizing

the cost/time associated with that decision.

The most popular such application involves the management of cash. The simplest version of a

cash-management scenario corresponds to the basic EOQ scenario, but with its inputs

reinterpreted. In particular: D is the rate at which cash is being accumulated from some business

activity into a noninterest bearing account (e.g., checking account); h is the interest being paid in

an interest-bearing account (e.g., money-market account), and is measured in dollars/(dollar x

time); and K is the fixed cost incurred in transferring any amount of money, Q, from the non-

interest to the interest-bearing account. Hence, Q* is the optimal transfer quantity.

Conclusion:

Consequently, the EOQ as a lot sizing tool is quite compatible with the principles of lean

systems. The EOQ and other lot-sizing methods answer the important question:

How much should we order?

Another important question that needs an answer is:

When should we place the order?

An inventory control system responds to both questions.

In selecting an inventory control system for a particular application, the nature of the demands

imposed on the inventory items is crucial. An important distinction between types of inventory is

whether an item is subject to dependent or independent demand. Retailers, such as JCPenney,

and distributors must manage independent demand items—that is, items for which demand is

influenced by market conditions and is not related to the inventory decisions for any other item

held in stock. Independent demand inventory includes

- Wholesale and retail merchandise

- Service support inventory, such as stamps and mailing labels for post offices, office sup¬plies

for law firms, and laboratory supplies for research universities

- Product and replacement-part distribution inventories

- Maintenance, repair, and operating (MRO) supplies—that is, items that do not become part of

the final service or product, such as employee uniforms, fuel, paint, and machine repair parts

Managing independent demand inventory can be tricky because demand is influenced by

external factors. For example, the owner of a bookstore may not be sure how many copies of the

latest best-seller novel customers will purchase during the coming month. As a result, the

manager may decide to stock extra copies as a safeguard. Independent demand, such as the

demand for various book titles, must be forecasted.