# Quantity Discount and EOQ

Quantity discount is a reduction in price offered by seller on orders of large quantities. Quantity discounts exist in different forms and in certain scenarios they may not be obvious. The well-known buy-1-get-1-free sale is actually a 50% quantity discount since you effectively purchase a unit at half the normal price.

Different forms of quantity discounts provide different purchase incentives to buyers. For example, the one discussed above has a tentency to compel the buyer to purchase more than they need at the moment i.e. the seller will not allow you to purchase just one unit at 50% of the full price. Another form of quantity discount which is based on the cumulative quantity purchased during a specific time period actually induces the buyer to continue purchasing from the current supplier and restricts switching to other suppliers.

## Implication for Decision Making

When purchasers following Economic Order Quantity (EOQ) model for ordering inventory have the opportunity to avail a quantity discount on order sizes greaters than their EOQ, they need to base their decision, apart from qualitative factors, on the net effect of the decision on the their income. A typical quantity discount has the following three effects on the income of a purchaser:

1. A saving in the form of reduced price
2. A saving in the form of reduced ordering costs
3. A loss in the form of increased total holding costs of inventory

A decision to avail the quantity discount should be taken only if the net effect of the above components on the income is positive.

The following problem tries to clarify decision making when there is an apportunity to obtain quantity discounts:

## Example

A retail store dealing in computer hardware imports an enterprise model solid-state drive (SSD) at a fixed price of $1000 per unit from the sole distributer of the SSD. On January 1, 2014, the store received an offer of 15% discount on orders of 300 or more units. Provided further that the estimated sales for the year are 600 units, the cost incurred per order is$1000 and the average holding cost per unit per annum is estimated to be $120 per unit. The full price is likely to remain$1000 during the year. Please ignore opening and closing inventories, safety stock etc.

Solution:

$$\text{Economic Order Quantity}\ (\text{EOQ}) \\ = \sqrt{\frac{\text{2}\times \text{Demand}\times \text{Order Cost}}{\text{Holding Cost per Unit}}} \\ = \sqrt{\frac{\text{2}\times \text{600}\times \text{1000}}{\text{120}}} \\ = \text{100 units}$$

Economic Order Quantity (EOQ)= 100 units

Discounted Order Quantity (DOQ) = 300 units

Annual Orders under EOQ = Demand ÷ EOQ = 600 ÷ 100 = 6 orders

Annual Orders under DOQ = Demand ÷ DOQ = 600 ÷ 300 = 2 orders

Average Inventory under EOQ = EOQ ÷ 2 = 100 ÷ 2 = 50 units

Average Inventory under DOQ = DOQ ÷ 2 = 300 ÷ 2 = 150 units

Saving from reduction in Price (A)
= Demand × Full Price × Discount Rate
= 600 × 1000 × 0.15 = 90,000

Saving from reduction in orders (B)
= Orders reduced × Order Cost
= (6 − 2) × 1000 = 4000

Increase in holding cost (C)
= Increase in average inventory × holding cost per unit per annum
= (150 − 50) ×120 = 12000

Net Effect on Income
= A + B − C
= 90000 + 4000 − 12000
= 82000

Since the net effect on income is positive, the store should avail the quantity discount.