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.
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.
The risk when using the EOQ is that ordering costs and lead times may be regarded as constant.
Within lean, the goal is to reduce lead times and setup times using methods such as SMED and Kanban.
At any time, optimal order size can be calculated, but when the optimal order size is 1, we have reached one-piece production, a final goal in lean manufacturing.
To determine the Economic Order Quantity, these costs must be analyzed further. Some assumptions are required:
We will use the following variables:
It is important to note which variables are annualized, which are per-order and which are per-unit.
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).
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 ) )
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.
1 Matching Supply with Demand: An Introduction to Operations Management
By Oskar Olofsson
Making lean work for you
World-Class / Lean Manufacturing
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