Impact of Imperfect Quality Items on Inventory Management for Two Warehouses with Shortages

Generally, the majority of the inventory models work on the concept that overall units produced must be perfect in terms of quality and that the storage capacity of the warehouse is unlimited. In fact, under realistic conditions, it is not possible to manufacture products with complete perfection. Furthermore, there are always some limits associated with storage capacity of the warehouse. This paper formulates an inventory model that considers the impact of imperfect quality items and shortages. The cost of storage in rented warehouse (RW) is greater than own warehouse (OW) due to fact there are better preservation facilities in RW. This work considers that defective items are completely withdrawn after the inspection process. The purpose of this inventory model is to establish the optimal order quantity and backorder size that maximize the total profit. Some numerical examples are solved, and a sensitivity analysis is included. KeywordsTwo warehouse, Imperfect quality items, Shortages.


Introduction
In the traditional economic order quantity (EOQ) inventory model there are two assumptions: i) the manufactured products are of good quality; and ii) all units are stored in one warehouse only. The perfection of products' quality is not always possible because it is affected straightaway by the reliability of manufacturing process and the handling system used during transportation. Furthermore, the availability of storage space along with the fixed budget to invest upon the raw materials are limited. Due to the restricted capacity of the own warehouse (OW) there is a need of an extra storage with better preserving facilities which has unlimited storage capacity. This extra storage facility is known as rented warehouse (RW). In this direction, some researchers have built inventory models by relaxing the two mentioned assumptions.
One inventory model with warehouse was proposed by Hartley (1976). Later, Sarma (1987) generalized Hartley (1976)'s inventory model by including the shipment cost from rented warehouse to own warehouse. Yang (2004) (2000) by imposing the assumption that in each cycle of the supply chain the shortages are backlogged and proved that backordering cost is inversely proportional to total profit. Eroglu and Ozdemir (2007) developed another extension of Salameh and Jaber (2000)'s inventory model in which the shortages are allowed. They suggested that there exists a quantity of items with good quality which can covert the existing demand as well as backorders. They concluded that an increment in the proportion of imperfect items in a lot leads to a decrement in total profit per unit time. Table 1 shows some research works related to inventory models.

Notation
The inventory model is built utilizing the following notation. ( ) probability density function of ( ) total sale revenue ($) ( , ) total cost ($) ( , ) total profit per cycle ($/unit of time) Decision variables:

Symbol
lot size (units) backorder quantity (units)

Modelling
At the beginning a lot size of units come into the inventory system at time = 0. It is assumed that from this lot size units are stored in the own warehouse (OW) and ( − − ) units are stored in the rented warehouse (RW).The rent warehouse has better preserving facilities than own warehouse which implies that the cost of holding the stock in rent warehouse is higher than that of own warehouse. The behavior of the inventory model is illustrated in Figure 1 and Figure 2 for two cases that can occur. Due to certain reasons such as improper transport, low labor skills, low quality of raw material, among others, the production process manufactures some goods that are not of high-quality. Due to this, a screening process must be conducted with the rate of units per unit time when the complete lot enters the inventory system. It is presumed that each lot contains an percent of imperfect items, where is a random variable whose probability density function is, ( ), and its mean is [ ] = . Thus, the lot has imperfect quality items and (1 − ) perfect items. The imperfect quality items are kept in store and sold at the termination of the screening phase ( ) at a salvage value of per unit, where ≤ . During time 1 all the demand is fulfilled from RW until the inventory level of RW reduces to zero. After, the upcoming demand is now covered from OW during time 2 . When the inventory level reduces to zero in OW, then the shortages occur during the time 3 . The time horizon ( ) of total inventory is given by The expected value of time horizon ( ) is Then, the whole revenue is defined as ( ) and the total cost is denoted as ( , ). The ( ) is the sum of sales of perfect items and the imperfect items, ( , ) = ordering cost + purchase cost + screening cost + holding cost + backordering cost The times , , 1 , 2 , and 3 (See Figure 1) are given by The total profit per unit of time, ( , ), is determined by the difference between total revenue per unit of time ( ) and total cost per unit of time ( , ) . Hence, Since is a random variable that follows a uniform distribution with known p.
Case I. When 1 ≥ , this case is illustrated by Figure 1.
With respect to Case II the total revenue ( ) is given as in Case I and the total cost ( , ) is given below Since, Then equation (14) is expressed as follows: Case II. When 1 < , this case is depicted in Figure 2.
The total profit per unit of time, ( , ), is given by Since is a random variable that follows a uniform distribution with known p.

Solution Procedure
The aim is to calculate the optimal order quantity * and optimal backorder quantity * . The necessary conditions for that the expected profit to be maximized are: If the solution ( * , * ) satisfies the conditions given by the equations (18), (19) and (20) prove that the function [ ( , )] is strictly concave with a negative-definite Hessian matrix. Therefore, the solution ( * , * ) is optimal. Thus, the solution procedure for each case is presented below. Step 3. Calculate the expected total profit [ ( , )] with equation (13).
Step 5. Report the optimal solution.
Algorithm for Case II. When 1 < Step 1. Calculate * and * by solving simultaneously equation (A6) and equation (A7) given in Appendix B.
Step 5. Report the optimal solution.

Numerical Examples
This section presents two numerical examples for illustrating the solution of two cases that can happen in the inventory model. Notice that 1 ≥ as it is expected.
The optimality test that ensures that the solution is optimal is as follows: Note that 1 < as it is expected.
The optimality test that guarantees that the solution is optimal is given below.

Sensitivity Analysis
The sensitivity analysis is executed for the example 1. The effects of fraction defective items ( Figure 3 reveals that the expected total profit decreases significantly as the fraction of imperfect items increases while the optimal order quantity and backorder quantity increase as the fraction of imperfect items increases (see Figure 4 and Figure 5).

Conclusion
This research work proposes a two-warehouse inventory model with shortages by seeing the impact of defective items. It was demonstrated as percentage of defective items increases then the expected total profit decreases. This inventory model extends the

Conflict of Interest
The authors confirm that there is no conflict of interest to declare for this publication.