Economic order quantity

Sci-Tech Dictionary: economic order quantity

(′ek·ə′näm·ik ′ör·dər ′kwän·ə·dē)

(industrial engineering) The number of orders required to fulfill the economic lot size.

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Investment Dictionary: Economic Order Quantity - EOQ

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An inventory-related equation that determines the optimum order quantity that a company should hold in its inventory given a set cost of production, demand rate and other variables. This is done to minimize variable inventory costs. The full equation is as follows:

where :
S = Setup costs
D = Demand rate
P = Production cost
I = Interest rate (considered an opportunity cost, so the risk-free rate can be used)

Investopedia Says:
The EOQ formula can be modified to determine production levels or order interval lengths, and is used by large corporations around the world, especially those with large supply chains and high variable costs per unit of production.

Despite the equation's relative simplicity by today's standards, it is still a core algorithm in the software packages that are sold to the largest companies in the world.

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Marketing Dictionary: economic order quantity

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Most economically efficient amount of materials to order at a given time. Economic factors such as volume discounts, inventory receiving and storage costs, usage rates, and risks associated with running out of stock, as well as the average time from order to delivery, are factored into the determination of optimal economic order quantity. Industrial marketers attempt to persuade buyers to order more frequently or in larger quantities, using a variety of techniques including volume discounts and installment payment options. See also operations research.

Business Dictionary: Economic Order Quantity (EOQ)

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Inventory decision model to calculate the optimum amount to order based on fixed costs of placing and receiving an order, carrying cost of inventory and the sales. EOQ is used both in manufacturing and retail inventory management.

Small Business Encyclopedia: Economic Order Quantity (EOQ)

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The Economic Order Quantity (EOQ) is the number of units that a company should add to inventory with each order to minimize the total costs of inventory—such as holding costs, order costs, and shortage costs. The EOQ is used as part of a continuous review inventory system, in which the level of inventory is monitored at all times, and a fixed quantity is ordered each time the inventory level reaches a specific reorder point. The EOQ provides a model for calculating the appropriate reorder point and the optimal reorder quantity to ensure the instantaneous replenishment of inventory with no shortages. It can be a valuable tool for small business owners who need to make decisions about how much inventory to keep on hand, how many items to order each time, and how often to reorder to incur the lowest possible costs.

The EOQ model assumes that demand is constant, and that inventory is depleted at a fixed rate until it reaches zero. At that point, a specific number of items arrive to return the inventory to its beginning level. Since the model assumes instantaneous replenishment, there are no inventory shortages or associated costs. Therefore, the cost of inventory under the EOQ model involves a tradeoff between inventory holding costs (the cost of storage, as well as the cost of tying up capital in inventory rather than investing it or using it for other purposes) and order costs (any fees associated with placing orders, such as delivery charges). Ordering a large amount at one time will increase a small business's holding costs, while making more frequent orders of fewer items will reduce holding costs but increase order costs. The EOQ model finds the quantity that minimizes the sum of these costs.

The basic EOQ formula is as follows:

TC = PD + HQ/2 + SD/Q

where TC is the total inventory cost per year, PD is the inventory purchase cost per year (price P multiplied by demand D in units per year), H is the holding cost, Q is the order quantity, and S is the order cost (in dollars per order). Breaking down the elements of the formula further, the yearly holding cost of inventory is H multiplied by the average number of units in inventory. Since the model assumes that inventory is depleted at a constant rate, the average number of units is equal to Q/2. The total order cost per year is S multiplied by the number of orders per year, which is equal to the annual demand divided by the number of orders, or D/Q. Finally, PD is constant, regardless of the order quantity.

Taking these factors into consideration, solving for the optimal order quantity gives a formula of:

HQ/2 = SD/Q, or Q = the square root of 2DS/H.

The latter formula can be used to find the EOQ. For example, say that a painter uses 10 gallons of paint per day at $5 per gallon, and works 350 days per year. Under this scenario, the painter's annual paint consumption (or demand) is 3,500 gallons. Also assume that the painter incurs holding costs of $3 per gallon per year, and order costs of $15 per order. In this case, the painter's optimal order quantity can be found as follows: EOQ the square root of (2 3,500 15) /3 187 gallons. The number of orders is equal to D/Q, or 3,500 / 187. Thus the painter should order 187 gallons about 19 times per year, or every three weeks or so, in order to minimize his inventory costs.

The EOQ will sometimes change as a result of quantity discounts, which are provided by some suppliers as an incentive for customers to place larger orders. For example, a certain supplier may charge $20 per unit on orders of less than 100 units and only $18 per unit on orders over 100 units. To determine whether it makes sense to take advantage of a quantity discount when reordering inventory, a small business owner must compute the EOQ using the formula (Q the square root of 2DS/H), compute the total cost of inventory for the EOQ and for all price break points above it, and then select the order quantity that provides the minimum total cost.

For example, say that the painter can order 200 gallons or more for $4.75 per gallon, with all other factors in the computation remaining the same. He must compare the total costs of taking this approach to the total costs under the EOQ. Using the total cost formula outlined above, the painter would find TC PD HQ/2 SD/Q (5 3,500) (3 187)/2 + (15 3,500)/187 $18,062 for the EOQ. Ordering the higher quantity and receiving the price discount would yield a total cost of (4.75 3,500) (3 200)/2 (15 3,500)/200 $17,187. In other words, the painter can save $875 per year by taking advantage of the price break and making 17.5 orders per year of 200 units each.

Overview

Assume that the demand for a product is constant over the year and that each new order is delivered in full when the inventory reaches zero. There is a fixed cost charged for each order placed, regardless of the number of units ordered. There is also a holding or storage cost for each unit held in storage (sometimes expressed as a percentage of the purchase cost of the item).

We want to determine the optimal number of units of the product to order so that we minimize the total cost associated with the purchase, delivery and storage of the product

The required parameters to the solution are the total demand for the year, the purchase cost for each item, the fixed cost to place the order and the storage cost for each item per year. Note that the number of times an order is placed will also affect the total cost, however, this number can be determined from the other parameters

The ordering cost is constant.
The rate of demand is constant
The lead time is fixed
The purchase price of the item is constant i.e no discount is available
The replenishment is made instantaneously, the whole batch is delivered at once.

EOQ is the quantity to order, so that ordering cost + carrying cost finds its minimum. (A common misunderstanding is that formula tries to find when these are equal.)

Variables

$Q$ = order quantity
$Q *$ = optimal order quantity
$D$ = annual demand quantity of the product
$P$ = purchase cost per unit
$C$ = fixed cost per order (not per unit, in addition to unit cost)
$H$ = 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)

At EOQ Ordering Cost And Carrying Cost Are Same.....

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 C, and we need to order D/Q times per year. This is C × D/Q

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

$TC = PD + {\frac{CD}{Q}} + {\frac{HQ}{2}}$ .

To determine the minimum point of the total cost curve, set its derivative equal to zero:

${\frac{dTC(Q)}{dQ}} = {\frac{d}{dQ}}\left(PD + {\frac{CD}{Q}} + {\frac{HQ}{2}}\right)=0$ .

The result of this derivation is:

$-{\frac{CD}{Q^2}} + {\frac{H}{2}}=0$ .

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

${\frac{H}{2}}={\frac{CD}{Q^2}}$

$Q^2={\frac{2CD}{H}}$

Therefore: $Q^* = \sqrt{\frac{2CD}{H}}$ .

Note that interestingly, Q* is independent of P, it is a function of only C, 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.

References

^ Hax, AC and Candea, D. (1984), Production and Operations Management, Prentice-Hall, Englewood Cliffs, NJ, pp. 135, http://catalogue.nla.gov.au/Record/772207

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	Economic Production Run Size Model (in accounting)
	Quantitative Methods (Models) (in accounting)
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