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Top1. Introduction
Inventory is an essential part of any firm as it constitutes the majority of the investment expenditure. So, inventory management plays an important role to operate a firm in a flexible manner. There are several functions that affect a firm like customer-demand, item deterioration, manufacturing defects, inventory shortage and also quantity-based discounts offered by the supplier. Management of these function can directly affect the total profit of retailer.
Inventory consists of raw material, partially complete and finished goods. So, for these types of inventories, there are ordering costs, holding costs and shortage costs associated with them. Purchasing cost may also involve inventory cost but most of the models assumed it as constant unless there is quantity discount involved. The quantity and time of placing the order are the two most important decisions in inventory management. These decisions will help a firm to increase total profit.
It is a common practice for a supplier to offer a discount on a large lot size. There are several reasons, why a supplier might choose to offer a retailer a discount. These reasons include the following: to increase cash flow, to decrease inventory levels of the products, to boost market share, or simply to retain retailers (Papachristos & Skouri, 2003). For retailers, ordering costs can be reduced on large quantity orders meanwhile holding costs will increase. These costs can manage through mathematical models so that retailers can earn more profit.
It is assumed that items are produced perfect during manufacturing but practically this is not possible due to some breakdown by a machine when running for a long time.(Sarkar, 2012a). So, it is necessary for the retailer to inspect imperfect items and separate imperfect and perfect items.
Furthermore, a retailer has a limited space to keep an inventory. In this situation, the retailer needs another space, named rent-warehouse, to keep inventory. Rent-warehouse has unlimited storing space with better-preserving facilities (Bhunia et al., 2014). Hence, rent-warehouse has higher holding costs than own-warehouse. As demand increases, shortages may occur. Also, imperfection of items could be the cause of shortages. (Aastha. et al.,2020, Priyanka & Pareek, 2020, Taha, 2017).
This paper deals with a company that sell items in a multi-warehouse environment with deterministic demand. Here, we assume all-units discounts, imperfect items and shortages. The purpose of this study is to optimize quantity and backorder items that maximize the total profit of the inventory. This model becomes more realistic by considering quantity discounts. This is an important feature for small retailers. Here, we use a procedure to obtain optimal solution and numerical examples are also provided to validate the results. To study the effect of parameters on the optimal solution, sensitivity analysis is also carried out.
The organization of this paper is as follows. Notation and assumption are introduced in the next section of this paper and then model formulation is defined in the following section. Based on the formulation, a solution procedure is mentioned in section 5 to find the optimal solution. In section 6, a numerical example is provided along with its optimal solution. Sensitivity analysis on different parameters is provided in section 7 and the conclusion is explained in the last section.