Joint replenishment with imperfect items and price discount

• Joint replenishment of multiple items with imperfect quality is studied.
• Two different models (with and without price discount) are investigated.
• Defective units affect the order quantity and the time of placing an order.
• Defective units also influence on the total expected cost.

A joint replenishment problem is presented to determine the ordering policy for multiple items having a certain percentage of defective units. The purpose of this paper is to study the impact of the percentage of defective units on the ordering policy. Two different scenarios are presented for joint replenishment problem: (1) without price discount and (2) with price discount. For each scenario, the total expected cost per unit time is derived and algorithms are presented to determine the family cycle length and the integer number of intervals that the replenishment quantity of each item will last. Numerical examples are presented and the results are discussed.