Make your own free website on


Graphical Solutions

Home | What is Economic Order Quantity? | Its Cost Components | EOQ Model with Purchases | Assumptions of the Model | Optimizing Economic Order Quantity | Graphical Solutions | Sample Problems | Answers

Graphical Solution


Graphical Solution

     If we minimize the sum of the ordering and carrying costs, we are also minimizing the total costs. To help visualize this we can graph the ordering cost and the holding cost as shown in the chart above.
     This chart shows costs on the vertical axis or Y axis and the order quantity on the horizontal or X axis. The straight line which commences at the origin is the carrying cost curve, the total cost of carrying units of inventory. As expected, as we order more on the X axis, the carrying cost line increases in a proportionate manner. The downward sloping curve which commences high on the Y axis and decreases as it approaches the X axis and moves to the right is the ordering cost curve. This curve represents the total ordering cost depending on the size of the order quantity. Obviously the ordering cost will decrease as the order quantity is increased thereby causing there to be fewer orders which need to be made in any particular period of time.

     The point at which these two curves intersect is the same point which is the minimum of the curve which represents the total cost for the inventory system. Thus the sum of the carrying cost curve and the ordering cost curve is represented by the total cost curve and the minimum point of the total cost curve corresponds to the same point where the carrying cost curve and the ordering cost curve intersect.

How to Calculate

Basic EOQ:

The objective is to determine the quantity to order which minimizes the total annual inventory management cost.

Thus: Minimize! Total cost per period = inventory holding costs per period + order costs per period

where Order Cost = The Number of Orders Placed in the period x Order Costs

and Carrying Cost = Average Inventory Level x the Carrying Costs of 1 unit of Stock for one period


         Q = order quantity

         A = demand per time period (e.g. Annual Demand)

         S = Carrying / Holding Cost of 1 unit of Stock for one period

         P = Order Cost

and the derivation set to zero we get the following formula:


      So we can see that the two cost elements at the economic order quantity are equal, one to the other; (compare with the graphical solution!)
If we now isolate the Q, we get the following Basic EOQ-Formula:

EOQ = Square root of 2AxP over S

Production EOQ:

Instead of instantaneous replenishment, we include the finite Production Rate R which leads to the following formula: (You can see, that production rate must be greater than demand rate, in order to fulfill the demand!)

EOQ = sqrt ( 2 * A * P / (S*(1-A/R))

 Backlogging EOQ:

By including the Backlogging Cost B, which is the cost of backlogging one unit per period, we get the following formula:

EOQ = sqrt (2 * A * P * (S+B) / S * B)


Economic Order Quantity

Economic  Order Quantity (EOQ), is an inventory model that attempts to minimize total inventory cost by answering the following two questions.

How much should I order? ( Economic Order Quantity )

How often should I place each order? ( Cycle Time )

This model assumes that the demand equation faced by the firm is linear.  In other words, the rate of demand is constant or at least nearly constant.  That the purchase price of the product or the cost is not a function of the number items delivered at any given time but determine based on anticipated demand and a price arranged between purchaser and supplier based upon an agreement the anticipated number of units to meet the demand over the coming period, typically annually.

The goal is to minimize total inventory cost.  Inventory cost are made up of holding and ordering cost. Holding cost include the cost of financing the inventory along with the cost of physically maintaining the inventory.  These costs are usually expressed as a percentage of the value of the inventory.  Ordering cost include the cost associated with actually placing the order.  These include a labor cost as well as a material and overhead cost.  The equation for total inventory cost is developed as follows:

Total Inventory Cost (TIC) = Holding Cost + Ordering Cost

TIC = (Average Inventory)(Holding cost per unit) + (Number of orders per year)(Ordering cost per order)


                 1                            D  

TIC = ———   QCH    +  ——— Co

                2                              Q


Where Ch = holding cost per unit; Co = ordering cost per order; D = annual quantity demanded; and Q = units per order. Using differential calculus techniques it is possible to show that total inventory cost are minimized when each order is made for the economic order quantity.   Economic order quantity is the number of units per order at which total inventory cost is minimized as shown in the following equation.


back to top