D Ordering cost: This is the cost of placing orders: each order has a fixed cost K, and we need to order D/Q times per year. This is K × D/Q Holding cost: the average quantity in stock (between fully replenished and empty) is Q/2, so this cost is h × Q/2 . To determine the minimum point of the total cost curve, calculate the derivative of the total cost with respect to Q (assume all other variables are constant) and set it equal to 0: Solving for Q gives Q* (the optimal order quantity): Therefore: Q* is independent of P; it is a function of only K, D, h. The optimal value Q* may also be found by recognizing that where the non-negative quadratic term disappears for which provides the cost minimum Example annual requirement quantity (D) = 10000 units Cost per order (K) = 40 Cost per unit (P)= 50 Yearly carrying cost per unit = 4 Market interest = 2% Economic order quantity = = 400 units Number of orders per year (based on EOQ) Total cost Total cost If we check the total cost for any order quantity other than 400(=EOQ), we will see that the cost is higher. For instance, supposing 500 units per order, then Total cost Similarly, if we choose 300 for the order quantity, then Total cost This illustrates that the economic order quantity is always in the best interests of the firm. Extensions of the EOQ model Quantity discounts An important extension to the EOQ model is to accommodate quantity discounts. There are two main types of quantity discounts: (1) all-units and (2) incremental. Here is a numerical example: Incremental unit discount: Units 1–100 cost $30 each; Units 101–199 cost $28 each; Units 200 and up cost $26 each. So when 150 units are ordered, the total cost is $30*100 + $28*50. All units discount: an order of 1–1000 units costs $50 each; an order of 1001–5000 units costs $45 each; an order of more than 5000 units costs $40 each. So when 1500 units are ordered, the total cost is $45*1500. In order to find the optimal order quantity under different quantity discount schemes, one should use algorithms; these algorithms are developed under the assumption that the EOQ policy is still optimal with quantity discounts. Perera et al. (2017) establish this optimality and fully characterize the (s,S) optimality within the