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Inventory control problem becomes one of the most crucial parts in a company. Basically, the problem is to determine the optimal time and quantity of replenishment. But, there are many factors affected the decision such as demand uncertainty, lifetime of the product, deterioration, unit sales price and discount offered by the supplier. In this paper, we consider a mathematical model for inventory control problem. We consider a multi-item problem with inventory and price-dependent demand and all-units discount offered by the supplier. In dealing with multi-item problem, there are two replenishment policies to consider, individual order and joint order policy. Numerical examples are given with different inventory and price-dependent demand functions for each item and compare the individual and joint order policy. We found that in our numerical example, joint replenishment policy gives lower total inventory cost compared to the individual order policy. [ABSTRACT FROM AUTHOR]

Good and properly organized inventory management is essentially needed for a company in order to ensure that its production process runs smoothly. There are many factors to be considered since they can distract the inventory management such as demand behavior, the life time of goods, uncertain replenishment time, the political and economic situation, and unexpected natural disaster. There are certain risks involved with bulk replenishment such as the higher holding cost and the possibility of damage to the stored goods. On the other hand, if we order too little, the risks lie in the possibility that the production process will stop, resulting in lost sales due to the unavailability of goods. One strategy to overcome this unavailability of goods is backorder, where the customers will wait until the next replenishment to obtain their goods. This paper develops a mathematical model for inventory-dependent demand with the backorder policy applied. The decisions variables in our model are the optimal order quantity and the number of backorders, which minimize the total inventory cost. We also performed a sensitivity analysis of the optimal solution and found that the decreasing rate of the inventory is affected by the autonomous demand (α) and the increasing demand factor (β). Inventory will deplete faster as α and β increase. The number of backorders is affected by the high demand when inventory reaches zero and backorders entail a certain cost. When demand is higher and the backorder cost is relatively cheap, the number of backorders becomes higher. [ABSTRACT FROM AUTHOR]