# Quantity Discounts

A quantity discount is a price discount on an item if predetermined numbers of units are ordered. In the back of a magazine you might see an advertisement for a firm stating that it will produce a coffee mug (or hat) with a company or organizational logo on it, and the price will be \$5 per mug if you purchase 100, \$4 per mug if you purchase 200, or \$3 per mug if you purchase 500 or more. Many manufacturing companies receive price discounts for ordering materials and supplies in high volume, and retail stores receive price discounts for ordering merchandise in large quantities.

The basic EOQ model can be used to determine the optimal order size with quantity discounts; however, the application of the model is slightly altered. The total inventory cost function must now include the purchase price of the item being ordered:

Purchase price was not considered as part of our basic EOQ formulation earlier because it had no impact on the optimal order size. In the preceding formula PD is a constant value that would not alter the basic shape of the total cost curve; that is, the minimum point on the cost curve would still be at the same location, corresponding to the same value of Q. Thus, the optimal order size is the same no matter what the purchase price is. However, when a discount price is available, it is associated with a specific order size, which may be different from the optimal order size, and the customer must evaluate the trade-off between possibly higher carrying costs with the discount quantity versus EOQ cost. As a result, the purchase price does affect the order-size decision when a discount is available.

## Quantity Discounts with Constant Carrying Cost

The EOQ cost model with constant carrying costs for a pricing schedule with two discounts, d1 and d2, is illustrated in Figure 12.4 for the following discounts:

Notice in Figure 12.4 that the optimal order size, Qopt, is the same regardless of the discount price. Although the total cost curve decreases with each discount in price (i.e., d1 and d2), since ordering and carrying cost are constant, the optimal order size, Qopt, does not change.

The graph in Figure 12.4 reflects the composition of the total cost curve resulting from the discounts kicking in at two successively higher order quantities. The first segment of the total cost curve (with no discount) is valid only up to 99 units ordered. Beyond that quantity, the total cost curve (represented by the topmost dashed line) is meaningless because above 100 units there is a discount (d1). Between 100 and 199 units the total cost drops down to the middle curve. This middle-level cost curve is valid only up to 199 units because at 200 units there is another, lower discount (d2). So the total cost curve has two discrete steps, starting with the original total cost curve, dropping down to the next level cost curve for the first discount, and finally dropping to the third-level cost curve for the final discount.

Notice that the optimal order size, Qopt, is feasible only for the middle level of the total cost curve, TC(d1)--it does not coincide with the top level of the cost curve, TC, or the lowest level, TC(d2). If the optimal EOQ order size had coincided with the lowest level of the total cost curve, it would have been the optimal order size for the entire discount price schedule. Since it does not coincide with the lowest level of the total cost curve, the total cost with Qopt must be compared to the lower-level total cost using Q(d2) to see which results in the minimum total cost.

 EXAMPLE12.4 A Quantity Discount with Constant Carrying Cost

Comptek Computers wants to reduce a large stock of PCs it is discontinuing. It has offered the University Bookstore at Tech a quantity discount pricing schedule, as follows:

The annual carrying cost for the bookstore for a PC is \$190, the ordering cost is \$2,500, and annual demand for this particular model is estimated to be 200 units. The bookstore wants to determine if it should take advantage of this discount or order the basic EOQ order size.

SOLUTION:

First determine the optimal order size and total cost with the basic EOQ model.

Although we will use Qopt = 72.5 in the subsequent computations, realistically the order size would be 73 computers. This order size is eligible for the first discount of \$1,100; therefore, this price is used to compute total cost:

Since there is a discount for a larger order size than 50 units (i.e., there is a lower cost curve), this total cost of \$233,784 must be compared with total cost with an order size of 90 and a discounted price of \$900:

Since this total cost is lower (\$194,105 < \$233,784), the maximum discount price should be taken, and 90 units should be ordered. We know that there is no order size larger than 90 that would result in a lower cost, since the minimum point on this total cost curve has already been determined to be 73.

## Quantity Discount Model Solution with POM for Windows

POM for Windows also has the capability to perform EOQ analysis with quantity discounts. Exhibit 12.4 shows the solution screen for Example 12.4. Exhibit 12.5 is a graph of the quantity discount model for this example generated by POM for Windows.

12-13. How must the application of the basic EOQ model be altered in order to reflect quantity discounts?

12-14. Why do the basic EOQ model variations not include the price of an item?