Economic Order Quantity for Items with Two Types of Imperfect Quality

The classical economic order quantity (EOQ) considers that the ordered items are of perfect quality. In this research, a model for the economic order quantity of imperfect quality items is developed, where the incoming lot has fractions of scrap and re-workable items. These fractions are considered to be random variables with known probability density functions. The demand is satisfied from perfect items and reworked items; whereas the scrap items are sold in a single batch at the end of the cycle with a salvage cost. A numerical analysis is provided to illustrate the sensitivity of the model to the fractions of scrap and reworked items.


Introduction
Although the economic order quantity is an old topic for research, it still is of wide interest to many industries today.This is because of the weakness of the developed economic order models that are based on unrealistic assumptions.This has led to the search for more realistic EOQ models which represent actual and real life cases.The main assumption in classical order quantity is that all items received are perfect; this particular assumption has been challenged by researchers.
Lee and Rosenblatt [1] have studied the effects of an imperfect production process on the optimal production cycle time.Cheng [2] has proposed an EOQ model with a demand dependent unit production cost and imperfect production processes.They formulate their inventory decision problem as a geometric program and solve the model to obtain closed form optimal solutions.Khouja [3] has extended the economic production quantity model to cases where production rate is a decision variable so that the unit production cost is a function of the production rate.Chang [4] has investigated the inventory problem for items received with imperfect quality, where, upon the arrival of the ordered lot, 100% inspection process is performed and the items of imperfect quality are sold as a single batch at a discounted price, prior to receiving the next shipment.Chiu et al. [5] have considered the economic production quantity model with a random defective rate and an imperfect rework process.Rezaei [6] has extended the traditional EPQ/EOQ model by accounting for imperfect quality items when using the backorder EPQ/EOQ formulae.An integrated EPQ model with imperfect quality and machine failure is proposed by Das et al. [7].Yassine et al. [8] have developed an EPQ model with disaggregation and consolidation of imperfect quality shipments.has developed an EOQ model when a single acceptance sampling plan with destructive testing and inspection errors is adopted.It is assumed that when the lot is rejected, items in the rejected lot are sold at a secondary market at a reduced price and the buyer will place another order.
Eroglu and Ozdemir [10] have developed an EOQ model in which each incoming lot contains some defective items and shortages are backordered.They assume that each lot goes through a 100% inspection to separate good from defective items; the defective items are classified as imperfect quality and scrap items.At the end of screening process, imperfect-quality items are sold as a single lot and scrap items are disposed of from inventory with disposal cost.Wee et al. [11] have developed an optimal inventory model for items with imperfect quality and shortage backordered, where poor-quality items exist during production.Poor quality items are picked up during the screening process and are withdrawn from stock instantaneously.Chen et al.
[12] have proposed a fuzzy economic production quantity model with imperfect products that can be sold at a discount price.Yoo et al. [13] have proposed a profit-maximization economic production quantity model that incorporates both imperfect production quality and two-way imperfect inspection.Lin [14] has developed an inventory model for items with imperfect quality and quantity discounts where buyer has exerted power over its supplier.Ho et al [15] integrate the vendor and the buyer in a model, in which, the order lot contains a random proportion of defective items and partial backlogging is allowed.
Salameh and Jaber [16] have hypothesized a production/inventory situation where items, received or produced, are not of perfect quality.They consider the issue that poor quality items are sold as a single batch by the end of the 100% inspection process.Hayek and Salameh [17] have studied the effect of imperfect quality items on the finite production model.When production stops, defective items are assumed to be reworked at a constant rate.And the optimal operating policy that minimizes the total inventory cost per unit time for the finite production model under the effect of imperfect quality is derived where shortages are allowed and backordered.Papachristos and Konstantaras [18] have examined models without shortages, probabilistic proportional imperfect quality, and withdrawing at the end of the planning horizon.Maddah and Jaber [19] have rectified a flaw in an economic order quantity model with unreliable supply, characterized by a random fraction of imperfect quality items and a screening process, developed by Salameh and Jaber [16].Then, they have analyzed the effect of screening speed and variability of the supply process on the order quantity.In addition, they extend the model by allowing for several batches of imperfect quality items to be consolidated and shipped in one lot.Jaber et al. [20] have extended the work of Salameh and Jaber [16] by assuming the percentage defective per lot reduces according to a learning curve.Khan et al. [21] have extended Salameh and Jaber's work for the case where there is learning in inspection.They consider situations of lost sales and backorders.Khan et al. [22] have used approach similar to Salameh and Jaber [16] to produce an optimal production/order quantity that takes care of imperfect processes, where the inspector may commit errors while screening.
In this article, we develop an inventory model where the incoming lot has imperfect quality items, either scrap or re-workable.When the lot of size is received, it is subjected to 100% inspection with a constant inspection rate.Inspection takes time .After screening, items will be classified in one of the following types: scrap, re-workable, or good.We assume that the probability density functions of the fractions of the scrap and re-workable items ( and , respectively) are known.The defective items are sold in a secondary market with a salvage cost, and re-workable items are returned to the supplier to be reworked and received back as good within the cycle of inspection.We assume the rework process is error free.
The article is organized as follows: Section 2 describes the model.Section 3 develops the mathematical model.Section 4 presents the numerical example.Section 5 discusses the sensitivity analysis.The article is concluded by Section 6.

Model Description
Figure 1 represents the model where the lot of size is received with purchasing price of per unit and the ordering cost of .It is assumed that each order contains a fraction of scrap and re-workable items and with known probability density functions and , respectively.Good and reworked items can be sold at per unit.On the other hand, scrap items will be sold in a batch at the end of the cycle with salvage (discount) per unit.The optimal order quantity is found by taking the difference between the total revenue and total cost, the latter of which consists of four types: procurement cost, inspection cost, rework cost, and inventory carrying cost.Revenues come from selling of good items and scrap items.The main assumptions of this model are: i. Shortages are not allowed.ii.Inspection and rework processes are error free.iii.The quantity of good items is sufficient to satisfy the demand during period of inspection.

Mathematical Model
Since shortage is not allowed, to avoid shortage, the number of good items is at least equal to the demand during inspection time: (1) By substitution of the value of , we get: (2) where .
Since are coming from a probability density functions (p.d.f), they will be limited as the following expression: The time needed to inspect the lot is: (4) The inventory level just before the end of the inspection process is: (5) The time (rework time) between the time the re-workable items are sent and the time they are received back is: (6) Inventory level after the removal of the scrap items and the return of the reworked items is: Inventory level just before receiving the reworked items is: (8) Inventory level when the reworked items are put in the inventory is: Finally, the time to consume is: (10) The expected total revenue is the summation of sales of the good items and scrap items and it is given as: (11) The expected total comprises four different costs.The first cost is the procurement cost: The rework cost is: The inspection cost is: And the expected inventory holding cost is: The expression for the expected total cost is: The expected total profit equals the expected total revenues minus the expected total cost: The expected cycle period is given by: (18) The expected total profit per unit time is: To find the optimal order quantity, the first derivative of Eq. ( 19) is taken, set to zero, and solved for : (20) From Eq. ( 20), we find the expression of the economic order quantity : (21) The second derivative is equal to: Since the second derivative is always negative under all values of , this means that there exists a unique value of that maximizes Eq. ( 19).
Note that when and , Eq. ( 21) reduces to the classical formula of : (23)

Numerical Example
From the expression of the expected total profit, there are several terms in that are not functions of .Therefore, these terms will be dropped from Eq. ( 21); and the new objective function will be in terms of minimizing the expected (relevant) cost per unit time: We assume the operation of the inventory model operates 8 hours a day, for 365 days a year, so the annual inspection rate is .Also, we assume scrap and re-workable fractions, and , are uniformly distributed with p.d.f as the following: (25) ( 26) The expected values in the model when the (p.d.f) is a uniform distribution are: Since probabilities always positive and : , Then, the optimal value of that optimizes the expected total profit per unit time is given by: units, with optimal expected cost per unit time = $ 7,395 From the classical EOQ model, the optimal value of , which is given by Eq. ( 23), is: = 1,414 Units, with expected cost per unit time = $ 7,424

Sensitivity Analysis:
In this section, we will study the effect of the variation in the fractions of the scrap and reworkable items on .We assume that the scrap and re-workable fractions are uniformly distributed with ~U (0, ) and ~U (0, ), respectively.
The results are generated and tabulated as follows: 1.
values with different fractions of scrap and re-workable items are shown in Table 1. 2. The ratio of to with different fractions of scrap and re-workable items are shown in Table 2. 3. The expected cost using with different fractions of scrap and reworkable items are shown in Table 3. 4. The expected cost using with different fractions of scrap and reworkable items are shown in Table 4. 5.The penalty for the deviation from , calculated by , with different fractions of scrap and reworkable items are shown in Table 5.
From Table 1, it is shown that increases with the variation in ; for instance, if we fix , then when , and when , .Also, increases with the variation in .For , the optimal pairs are and .3, it is shown that increases with the variation in ; for instance, if we fix , then when , ; and when , .On the other hand, decreases with the variation in .For , the optimal pairs are and .

And from Table
It's also shown that the developed economic order quantity ( ) is greater than the classical economic order quality ( ) under the same variation in and , which is illustrated in Table 2 by the ratio being greater than one.Finally, the expected cost for is always less than that for under the same variation in and as shown in Tables 3 and 4.

Conclusion
The economic order quantity is developed for the inventory model with imperfect quality items.The model proposed here considers that the incoming lot has a fraction of scrap and reworkable items, and the lot will go through a 100% inspection.The scrap items will be sold in a secondary market and re-workable items will be returned back to the supplier and received again within the same cycle as good items.Furthermore, an expression for the optimal lot size has been developed.The optimal lot size is affected by the fraction of imperfect quality items; the lot size increases with the increase in the fraction of scrap items and re-workable items.Also the expected total profit decreases with the fraction of scrap and re-workable items.

Table 1 .
with different variation of scrap and re-workable items.

Table 2 .
The ratio of to with different variation for scrap and re-workable items.

Table 3 .
Expected cost using with different variation for scrap and re-workable items.

Table 4 .
Expected cost using with different variation for scrap and re-workable items.

Table 5 .
The penalty for deviation from with different variation for scrap and re-workable items