Wilson Inventory Calculator

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What does Economic Order Quantity mean?

Economic Order Quantity (EOQ) is the level of inventory that minimizes the total cost of holding and ordering inventory over a period of time. Usually the time period is one year.

What is the Total Cost of Inventory?

The total cost of inventory is the sum of the purchase, ordering and holding costs. As a formula:
TC = PC + OC + HC, where TC is the Total Cost; PC is Purchase Cost; OC is Ordering Cost; and HC is Holding Cost.

Assumptions

To determine the Economic Order Quantity, these costs must be analyzed further. Some assumptions are required:

Purchase Cost is a straight-forward “unit cost X number of units” calculation. In other words, volume discounts do not apply. As well, the unit cost remains constant over the year.
Order Cost is a fixed overhead cost, and remains constant over the year. It represents, for example, the time value for employees to write up an order, mail it, follow up, inspect the received goods, and make the payment.
Holding Cost is fixed over the year. It represents warehouse space, with services such as refrigeration or insurance or security. It also includes the interest cost, which should be set to the “risk-free opportunity-cost” rate.
The rate of demand is constant over the year.
The total quantity ordered is delivered in one batch.
The lead time (between placing an order and receiving it) is constant and does not depend on the order quantity.
The quantities are large enough that calculus may be used to determine a minimum point. Calculus requires smooth functions on the real number system. Order quantities assume integral units.

Variables

We will use the following variables:

Q = Quantity being ordered
Q* = the optimal order Quantity: the result being sought
D = annual Demand for the item, over the year
P = unit Purchase cost
O = cost of one Order, regardless of the number of units in the order

Sometimes shown as S, meaning the cost to Set-Up the next production run

H = annual cost to Hold one unit

It is important to note which variables are annualized, which are per-order and which are per-unit.

Do the easy math

Using the variables, here are the components of the first equation (TC = PC + OC + HC):

PC = P x D :                       Purchase Cost = unit Purchase cost times the annual Demand
OC = (D x O) / Q :             Order Cost = annual Demand times cost per Order,
divided by the order Quantity (number of units)
HC = (H x Q) / 2:               Holding Cost = annual unit Holding cost times order Quantity (number of units),
divided by 2 (because throughout the year, on average the warehouse is half full).

So TC = PC + OC + HC = (P x D) + ( (D x O) / Q) + ( (H x Q) / 2).

Use calculus to determine the minimum

To minimize TC for Q, determine the first derivative of this formula and solve for zero.

dTC(Q)/dQ = d ( (P x D) + ( (D x O) / Q) + ( (H x Q) / 2) )/dQ
= (H / 2) – (D x O) / ( Q2 ) )
= zero

To solve for Q*: (the optimal order Quantity):

(H / 2) = (D x O) / ( Q*2 ) )

Therefore Q*2 = 2 x (D x O) / H.

Thus Q* = the square root of 2 x (D x O) / H, and does not depend on the unit purchase cost.

In English: the optimal order Quantity is the square root of 2 times the annual Demand times the cost of one Order divided by the annual cost to Hold one unit.

References:

1 Matching Supply with Demand: An Introduction to Operations Management

2 Operations Management

Oskar Olofsson, 2009

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I am a Swedish-based Lean consultant, and the owner of the World-Class-Manufacturing.com web site.

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Currency
Demand per year
Setup costs/ fixed cost per order
Unit holding cost (warehouse, insurance etc.)
Lead time in days
Working days per year