Economic order quantity with partial backordering and sampling inspection
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Abstract
To access the efficient inventory system, managers should consider all the situations that have happened in reality. One of these situations is the presence of the defective items in each received lot and the other situation is being the group of customers that do not wait to fulfill their requirements from the vendor and choose another one to get their orders so the proportion of the backordered items becomes lost sales. In this paper we consider both mentioned situations simultaneously to model the inventory system while the proportion of backordering is constant and the imperfect rate follows a uniform distribution, also the particular sampling process is considered that is explained in detail in "Problem definition". Our purpose in this paper is to access the optimum value for the total revenue in a year by a particular solution method that is provided in "Solution method". After these sections we provide the numerical results in "Numerical result" to show the effect of sensitive parameters on the decision variables and the total profit.
Keywords
Inventory system Imperfect items Partial backordering InspectionIntroduction and literature review
Summary of the inventory articles that are works on defective items and partial backordering
Category  Authors  Remarks 

Models with defective items  Salameh and Jaber (2000)  Full inspection and the vendor sells defective products after the inspection process 
Papachristos and Konstantaras (2006)  Extension of the model that is provided by considering shortage  
Eroglu and Ozdemir (2007), Hsu and Hsu (2013a, b), Rezaei (2005), Wee et al. (2007)  Extension of the model that is provided by considering shortage and backordered items  
Hayek and Salameh (2001), Konstantaras et al. (2007), Shekarian et al. (2014)  Consideration of the situation that the defective items are reworked and sold as a perfect items with the same price  
Learning impacts on inspection  
Rezaei and Salimi (2012)  Consideration of the situation that the inspection is done by the supplier and the buyer pay more price instead of it  
Hsu and Hsu (2013), Hsu and Hsu (2013), Khan et al. (2014), Khan et al. (2011)  Errors that occurred in the process of the inspection  
Skouri et al. (2014)  Full inspection  
Partial backordering  Mak (1987)  Considers backordered fraction as an uncertain parameter 
Abad (1996, 2000, 2001, 2003, 2008), San José et al. (2007), S. Papachristos and Skouri (2000)  Dependence of backordered fraction on the replenish time  
Vrat and Padmanabhan (1990), Padmanabhan and Vrat (1995), Ouyang et al. (2003), Chu and Chung (2004), and Dye et al. (2006)  Dependence of backordered fraction on the backlog size  
Pentico and Drake (2009)  A survey of deterministic model for the EOQ and EPQ models with partial backordering 
Problem definition
There are some assumptions that should be considered to model the inventory system to access more efficient and usable results. One of these assumptions is inspection. There are some defective items in each order that a firm or enterprise receives, these items should be recognized and managers should choose the particular decision according to their inspection strategy. In this paper we consider an inspection approach that is explained in the following. In this approach we consider three levels for the number of defective items in each sample that is chose from each order randomly.
In the following, we define notations that are used in the associated model.
 D

Demand in a year
 \(c_{0}\)

Ordering cost
 \(c_{h}\)

Holding cost per unit per year
 \(c_{p}\)

Unit purchasing cost
 s

Unit selling price
 \(c_{b}\)

The cost related to a unit of backordered items in a year
 \(c_{g}\)

Goodwill cost related to a unit of lost items
 x

Inspection rate during a unit of time
 p

Defective rate
 d

Cost of inspecting a unit per unit of time
 θ

Number of defective products in a sample of n items
 n

Size of the sample
 r

Cost for returning a defective item
 R

The cost that is paid because of the wrong rejection by the vendor
 γ

Fraction of backordered shortage
 T

Ordering cycle duration
 φ

The fraction of demand that is fulfilled from stock
 TP

Total profit per cycle
 ETP

Expectation of total profit per cycle
 E(.)

Mathematical expectation
 f (p)

Function of probability density of defective rate
 E[p]

Expectation of defective rate
According to the mentioned explanations we have three situations regarding the number of defective items in each order. In the first situation, the number of defective items that are in the sample chosen from the lot is more than the upper line which is α _{2,} so the buyer rejects the lot and receives another order without any defective items. In this situation it is beneficial for the managers to return the lot to the supplier, because they conclude there are many defective items in each lot and it affects their prestige badly if these lots are delivered to their customer. In the second situation, the number of defective items is between α _{1} and α _{2} so the buyer inspects all the items because the mangers know that if the number of defective items in a sample becomes between two particular numbers it is beneficial that the inspection process is considered though it leads to inspection cost. The defective items are separated from the perfect items. Then perfect items are sold at the particular price within the cycle and the defective items are sold after the inspection time at the lower price than the perfect items. In the third situation the number of defective items that are in the sample is lower than the lower line, α _{1}, so the buyer does not intend to the inspection process and prefer to return the lot to the supplier. All the items are sold with the particular price that is determined for the perfect items. After selling the items customers can return the defective items and get the particular amount of money instead of it. Each of these situations has a particular probability that affects the formulation of the total revenue function. In the following, we explain these situations precisely.
The first situation: \(\theta > \alpha_{2}\)
The second situation: \(\alpha_{1} \le \theta \le \alpha_{2}\)
The third situation: \(\theta \le \alpha_{1}\)
In all these cases we consider two types of cost of shortage that has been explained before. In the following section, we provide the solution method according to the solution that has been provided by Pentico and Drake (2009).
Solution method
To access a general optimum order quantity we combine three cases so the expected for p is calculated in three cases as below:
First case: \(\theta > \alpha_{2}\)
Also in two next parts we have formulated this expectation for other situations mentioned before.
Second case: \(\alpha_{1} \le \theta \le \alpha_{2}\)
Third case: \(\theta \le \alpha_{1}\)
In this section we have obtained the optimum quantity for the decision variables and total profit. In the following section we provide the numerical result.
Computational and practical results
 a.
If the number of defective products \(\theta\) is more or equal to \(\alpha_{2}\) = 4, the buyer rejects the order and receives another lot instead of it that it does not have any defective items.
 b.
If the number of defective products \(\theta\) is equal to 2 or 3 all the items should be inspected.
 c.
If the number of defective products \(\theta\) is lower or equal to \(\alpha_{1}\) = 1 no item is inspected.
The probability of different situation
\(b \le 0.0600\)  \(0.0600 \le b \le 0.1500\)  \(0.1500 \le b \le 0.2500\)  

\(\theta \le 1.00\)  0.79600  0.18600  0.01800 
\(1.00 \le \theta \le 4.00\)  0.23700  0.46000  0.30400 
\(\theta \ge 4.00\)  0.00400  0.03200  0.96300 
Quantity of \(E_{3} [p]\), \(E_{2} [p]\), \(E_{3} [p]\) and E(p)
Row  E(p)  \(E_{3} [p]\)  \(E_{2} [p]\)  \(E_{1} [p]\)  ETP  Row  E(p)  \(E_{3} [p]\)  \(E_{2} [p]\)  \(E_{1} [p]\)  ETP 

1  0.01  0.031  0.112  0.196  680.87  11  0.11  0.047  0.118  0.197  1868.6 
2  0.02  0.039  0.114  0.196  716.74  12  0.12  0.045  0.091  0.120  2063.6 
3  0.03  0.047  0.116  0.197  753.51  13  0.13  0.045  0.096  0.129  1997.3 
4  0.04  0.034  0.086  0.195  1486.6  14  0.14  0.045  0.098  0.139  1972.0 
5  0.05  0.036  0.090  0.195  1527.3  15  0.15  0.046  0.101  0.149  1963.6 
6  0.06  0.038  0.095  0.195  1596.0  16  0.16  0.046  0.104  0.158  1940.6 
7  0.07  0.040  0.100  0.196  1660.8  17  0.17  0.046  0.107  0.168  1902.4 
8  0.08  0.042  0.104  0.196  1710.7  18  0.18  0.046  0.110  0.177  1876.3 
9  0.09  0.044  0.109  0.196  1769.7  19  0.19  0.046  0.113  0.187  1853.5 
10  0.10  0.046  0.113  0.197  1815.4  20  0.20  0.047  0.116  0.197  1831.1 
Quantity of \(T^{*}\) according to the different quantity of \(\gamma\)
\(\gamma\)  \(T^{*}\) 

0.1  0.1454 
0.2  0.0787 
0.3  0.559 
0.4  0.0441 
0.5  0.0366 
Quantity of \(\varphi^{*}\) according to the different quantity of \(\gamma\)
\(\gamma\)  \(\varphi^{*}\) 

0.1  0.74 
0.2  0.63 
0.3  0.49 
0.4  0.32 
0.5  0.23 
Quantity of ETP according to the different quantity of \(\gamma\)
\(\gamma\)  ETP 

0.1  75559 
0.2  28321 
0.3  7393.1 
0.4  1972.0 
0.5  514.72 
0.6  ETP < 0 
0.7  ETP < 0 
0.8  ETP < 0 
0.9  ETP < 0 
As we see in Figs. 4, 5 and 6, by increasing \(\gamma\) the optimum value of both decision variables φ and T, also expectation of total profit, ETP, decrease. Our purpose is to increase the expectation of the profit, so by considering the greater value for γ this purpose is accessible, though it should be noted that for this case these results are gained and absolutely for the other cases results could be changed. So according to considered cases, managers should set the sensitive parameters with the particular values to access the best results for their decision variables and expected value of the total profit. In this part, we provided some numerical results according to the data related to a diary store, analyze these results regarding the sensitive parameters to find out how this model can impact on the expectation of total profit and decision variables. It should be noted that we verify two conditions for the proof of the concavity of ETP (see “Appendix A”) to find out if it is concave or not according to the quantity of parameters, and it is found out that it is concave. Also this part is provided to find out how the provided model in this paper can improve the process. In reality, to access the precise process of inventory system and being successful among many companies, it is necessary to consider the assumptions that are provided in this paper to model the inventory system, partial backordering and particular process of inspection because of the defective items. Nowadays competition between the enterprises has been very intense and without such precise inventory model that includes many aspects of inventory system it is not possible to be successful. With the provided model in this paper we can make the customers more satisfied than before, also by considering the perfect strategy for inspection, managers can determine the decision variables perfectly. Additionally the cost of inventory system is determined more similar to the real world and the answers that are obtained from the provided model are more reliable to cope with the fluctuations of the price, amount, etc. Also responsiveness is an important factor that is considerable for the managers; with the provided model, this factor increases and absolutely customers are more satisfied. Additionally with this model managers can increase their system flexibility to respond to the changes and provide their certain services with the higher level, also demand and supply can be determined perfectly.
Conclusion
There are some assumptions that are really essential to be considered to model the inventory system. One assumption is that there are some defective items in each order that a firm or enterprise receives, these items should be recognized and managers decide what to do according to their inspection strategy. In this paper we consider an inspection approach that is explained in the following. In this approach we consider three levels of number of defective items in each sample that is chosen from each order randomly. According to the number of defective items that is in an order we decide what to do for with the order. According to the mentioned explanation we define three levels of defective items in each sample according to their numbers. If this number is less than α_{1} it is not necessary to inspect all the items, else if this number is between α_{1} and α_{2} all the items should be inspected and if this number is more than α_{2} the order is rejected and another order without any defective items is received. The rate of imperfect items in each order is p and regarding the number of defective items it has three levels, lower than p _{1}, between p _{1} and p _{2} and upper than p _{2}. Another assumption is related to the customers. There are two types of customers. The first one is the customers that do not change the vendor that they choose to fulfill their demand though they know that they should wait more than the regular time for their order. On the other hand the second one is the customers that do not be patient enough to wait to fulfill their demand; these customers prefer to receive their orders by the other vendors because they do not want to wait or their demand is critical so their demand should be fulfilled soon. Because of the second ones a particular proportion of backordered items become lost sales so we have partial backordering instead of full backordering. We have proposed the model by considering these assumptions and obtain the optimum quantity for the total profit and the decision variables. Next we have provided the numerical result to show how the sensitive parameter affects the total profit and decision variables, also how the different situations affect the total profit. This model is the first one that considers both partial backordering and the process of inspection simultaneously and is so helpful for the managers to cope with the fluctuations of the inventory system in the real world as we discussed in Sect. 5. All managers surely are interested to make the inventory system closer to the real world. There are some assumptions that make the model more complicated. For example, the demand rate and length of the cycle could be considered as stochastic variables, also the fraction of the backordered items could be provided by the particular distribution function to make the model more realistic. Although the selling price is considered as constant, it could be considered as a particular function of certain parameters.
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