EOQ setting under general cost structures. Design of optimal quantity discount schedules In presence of a strategic customer, who responds optimally to discount schedules, the design of an optimal quantity discount scheme by the supplier is complex and has to be done carefully. This is particularly so when the demand at the customer is itself uncertain. An interesting effect called the "reverse bullwhip" takes place where an increase in consumer demand uncertainty actually reduces order quantity uncertainty at the supplier. Backordering costs and multiple items Several extensions can be made to the EOQ model, including backordering costs and multiple items. In the case backorders are permitted, the inventory carrying costs per cycle are: where s is the number of backorders when order quantity Q is delivered and is the rate of demand. The backorder cost per cycle is: where and are backorder costs, , T being the cycle length and . The average annual variable cost is the sum of order costs, holding inventory costs and backorder costs: To minimize impose the partial derivatives equal to zero: Substituting the second equation into the first gives the following quadratic equation: If either s=0 or is optimal. In the first case the optimal lot is given by the classic EOQ formula, in the second case an order is never placed and minimum yearly cost is given by . If or is optimal, if then there shouldn't be any inventory system. If solving the preceding quadratic equation yields: If there are backorders the reorder point is: ; with m being the largest integer and μ the lead time demand. Additionally, the economic order interval can be determined from the EOQ and the economic production quantity model (which determines the optimal production quantity) can be determined in a similar fashion. A version of the model, the Baumol-Tobin model, has also been used to determine the money demand function, where a person's holdings of money balances can be seen in a way parallel to a firm's holdings of inventory. Malakooti (2013) has introduced the multi-criteria EOQ models where the criteria could be minimizing the total cost, Order quantity (inventory), and Shortages. A version taking the time-value of money into account was developed by Trippi and Lewin. Imperfect quality Another important extension of the EOQ model is to consider items with imperfect quality. Salameh and Jaber (2000) are the first to study the imperfect items in