James M. Cargal

Mathematics Department

Troy University – Montgomery Campus

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 (unfortunately many of the best books on the subject are out of print). 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. We will see that 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. Hence you might wonder how I can state that large corporations do use the EOQ formula. Let’s just say that I have good sources of information that I feel can be relied upon.

The Variables of the EOQ Problem

Let us assume that we are interested in optimal inventory policies for widgets. The EOQ formula uses four variables. They are:

D:

Q:

C:

h:

The demand for widgets in quantity per unit time. Demand can be thought of as a rate.

The order quantity. This is the variable we want to optimize. All the other variables are fixed quantities.

The order cost. This is the flat fee charged for making any order and is independent of Q.

Holding costs per widget per unit time. If we store x widgets for one unit of time, it costs us x@h.

The EOQ problem can be summarized as determining the order quantity Q, that balances the order cost C and the holding costs h to minimize total costs. The greater Q is, the less we will spend on orders, since we order less often. On the other hand, the greater Q is the more we spend on inventory. Note that the price of widgets is a variables that does not interest us. This is because we plan to meet the demand for widgets. Hence the value of Q has nothing to do with this quantity. If we put the price of widgets into our problem formulation, when we finally have finally solved the optimal value for Q, it will not involve this term.

The EOQ Inventory Formula by J. M. Cargal

The Assumptions of the EOQ Model

The underlying assumptions of the EOQ problem can be represented by Figure 1. The idea is that orders for widgets arrive instantly and all at once. Secondly, the demand for widgets is perfectly steady. Note that it is relatively easy to modify these assumptions; Hadley and Whitin [1963] cover many such cases. Despite the fact that many more elaborate models have been constructed for inventory problem the EOQ model is by far the most used.

Figure 1 The EOQ Process

An Incorrect Solution

Solving for the EOQ, that is the quantity that minimizes total costs, requires that we formulate what the costs are. The order period is the block of time between two consecutive orders. The length of the order period, which we will denote by P, is Q/D. For example, if the order quantity is 20 widgets and the rate of demand is five widgets per day, then the order period is 20/5, or four days. Let Tp be the total costs per order period. By definition, the order cost per order period will be C. During the order period the inventory will go steadily from Q, the order amount, to zero. Hence the average inventory is Q/2 and the inventory costs per period is the average cost, Q/2, times the length of the period, Q/D. Hence the total cost per period is:

Q2 h

TP C h

C

2D

2D

If we take the derivative of Tp with respect to Q and set it to zero, we get Q = 0. The problem is solved by the device of not ordering anything. This indeed...