Impact of Inspection Error of an Economic Production Quantity Model with Acceptance Sampling, Defective Items, and Rework

: In this model, the manufacturer needs to inspect and rework defective items due to the flawed manufacturing process. When checking for defective products, two types of errors, namely Type 1 error and Type 2 error, may occur at the inspection stage. If a lot of defective materials are rejected during the inspection, a non-destructive screening process separates the material into reworkable and non-defective categories. This study includes inspection rate insufficiency (IRI) and inspection rate sufficiency (IRS) based on the relationship between inspection and production rates. Rework Priority Policy (RPP) is followed in every scenario. This paper aims to minimize the inspection error and optimize the total profit. Numerical examples help demonstrate the value of this well-established model. A sensitivity analysis of this study was carried out to check its accuracy


Nomenclature a
Selling price per unit α Manufacturer inspection rate The time for defective items to be reworked by the manufacturer m 1 Length of the cycle in sufficient inspection m 2 Length of the cycle in insufficient inspection P r Production rate, where (P r > D) P q Production quantity with perfect quantity while manufacturing for delivery for the first lot q 1 to the retailer µ Defective rate Q Total production quantity per production run R w The manufacturer rework rate i L T The time for manufacturer's production run (for i = 1 IRI scenario, i = 2 IRS scenario)

Introduction
Various standard variables in the dielectric material, fabrication, fabrics, and petrochemical industries, such as raw material quality variation, internal material flaws, equipment failures, and working conditions, always impact the manufacturer's production process.As a consequence, Sobhani et al. [1] and Guha and Bose [2] discussed the manufacturer's flawed manufacturing process may result in the production of goods of inferior quality.To ensure that the market's needs can be met by flawless commodities, such defective things must be screened out.Regarding the inspection procedure, the producer should check all completed products during the manufacturing cycle.This requires a specific amount of resources, such as time, equipment, and a workforce (Ullah & Kang [3], Sarkar & Saren [4]).However, the inspection and rework processes are time-consuming, which can delay the manufacturer's on-time delivery and make meeting the retailer's order deadlines even more challenging.The Rosenblatt and Lee [5] study is one of the first to examine how flawed production processes affect economic production quantity (EPQ) and economic order quantity (EOQ).A factory's quality control rate may be lower than producibility because of a lack of manual labor or sophisticated test equipment, creating the IRI situation discussed by Zhou et al. [6].An EPQ model for a batch production facility and manufacturer capable of producing products of varying quality.The manufacturer has lots of goods, and systems are expected to be used to evaluate the quality of each batch of products.The manufacturer's inspection rate may play a variety of functions depending on the inspection conditions for the timely delivery of flawless goods to the store.This work aims to determine the implications of the observation rate under the IRI and IRS scenarios on the effective logistics strategy, which is thus the first goal of this work.Because the acceptance sampling procedure is imperfect, two different errors may occur.Type 1 errors occur when a lot is deemed unacceptable when it is acceptable.Lot acceptance errors of Type 2 occur when a lot is accepted when it should not have been.In particular, the notable contributions of this paper can be explained as follows: As a result of the relationship between the vendor's checking and fabrication rate, the probability of error occurs during inspection time when this study examines the entire beneficial production policy using two alternative scenarios, which include inspection rate insufficiency (IRI) and inspection rate sufficiency (IRS).
To maximize production and profit, we consider the manufacturer in this task.Items can be produced with imperfections.In the interim, time misclassification may also happen.Here, we choose the Rework Productivity Policy (RPP) for asynchronous rework production time.Using appropriate numerical examples and sensitivity analysis, we demonstrated the effect caused by changes in various parameters.To the best of our knowledge, no one has taken the impact of inspection error and rework for defective goods into consideration up to this point.As a result, our approach offers a fresh managerial perspective that enables a provider to maximize the system's total profit.We will discuss the problem description, the numerical analysis, and the conclusions in the following sections.

Literature review
More plausible scenarios are included in the traditional economic production scale model.Numerous scholars have already thought about more effective operational strategies that consider defective products based on EPQ, where the manufacturer is personally liable for quality assurance and whose inspection rate exceeds the demand rate.Al-Salamah [7] developed an EPQ model to examine the scenario involving destructive and accurate acceptance sampling methods and defective manufacturing and inspection.Cunha et al. [8] implemented an EPQ model that considers faulty items with a portion of back-orders, giving a discount for packages of substandard quality.Marchi et al. [9] used the EPQ model to solve the energy efficiency problem; the EPQ model incorporates manufacturing, accuracy, and dependability training with defective materials.Numerous academics have also underlined how important it is for the producers to be in charge of the checking accuracy procedures in the supply system.Su [10] A cashback rewards policy created a holistic inventory model that accounts for defective items and assumes that the manufacturer undertakes 100% scrutiny processes with minimal checking durations.Al Hanbali et al. [11] created a maintenance model including parts quality, lead-time, and inspection errors, and they increased the optimal cost consistency rate as average deterioration per hour.Chen [12] tried to improve the cost of the joint economic lot-size problem (JELP) model and operational strategies for damaged and perishable goods under RPP.In other words, the manufacturer decides whether to repair defective products, and the rework process must be fully finished before the shipment arrives (this situation can be considered a notable case of RPP).The above investigations were carried out in the IRS situation, where the supplier's performance rate sets the maximum limit of readily available flawless quality materials due to the inspector's adequate inspection rate -the IRI scenario.Given the situation, it is essential to determine how IRI affects capacity utilization and develop relevant integrated operational solutions for defective items.In addition, the manufacturer's production system may provide repair or rework facilities, allowing the company to directly apply rework policies for faulty items.Taleizadeh et al. [13] The manufacturer spends a lot of time on quality.The review and rework process will begin after the inspection process is inspected by Zhou et al. [6].Bazan et al. [14] invigilated that the manufacturer's inspection rate is not less than its manufacturing rate.The processing interval of a vendor can be viewed as equivalent to its analysis process, and the repair process can be carried out after the production process is complete, as finished goods can be inspected until they are manufactured.Bräuer and Buscher [15] discussed the implications of both the relaunch policies on current production policies and the preferences of inclusive warehouse management towards these policies, which are the second topic of this paper and will be addressed in further research on texts.Hsu and Hsu [16] and Yoo et al. [17] designed an inventory model where items are screened, with the potential for Type 1 and Type 2 errors to result in the misclassification of goods.Jaber and Khan [18] considered an economic order quantity model in which items are misclassified at fixed rates based on quality and imperfect screening imperfections.Zheng et al. [19] discussed that after describing the average deterioration rate, the likelihood ratio sequence, and the monotonicity of the average failure time in the average deterioration rate, the original acceptance test is used as an acceptance index to simplify the problem successfully.Cheng et al. [20] developed a new approach that allows consumers to examine the product's operational reliability indicators without needlessly rejecting unreliable samples, which provides a higher chance of product acceptance than traditional acceptance sampling schemes.Salameh and Jaber [21] developed the EPQ model, which takes into account goods with flaws that can be discovered through screening and combined into a single batch that will be sold after the production cycle.Wang et al. [22] explored the optimization of deterioration control limits and periodic inspection intervals and the effect of inspection error.
Malik et al. [23] discussed the essentials of setting up a good coordination framework to resolve SC disagreements.coordination in a two-member SC with a flexible production system under stochastic demand and buyer service level constraints.Kalantari and Taleizadeh [24] proposed an EPQ model for estimating the number of shipments, the size of replenishment lots, and the selling price with rework and multi-shipments into a single model.Chiu et al. [25] proposed that the EPQ model includes several deliveries and a known percentage of defective products.After the production process, the defective goods are either scrapped or repaired.Zhang et al. [26] stated that the impact of preliminary studies with two types of defects is considered in the composite design problem.The quality loss depends on equipment wear and tear and the state of the manufacturing process.Karthick and Uthayakumar [27] detail that defective products are exposed due to equipment malfunctions during production, and the buyer undergoes a quality inspection process to identify faulty products.Cárdenas-Barrón et al. [28] alleviate the existing studies with integer values for quantity of product and freight variety.In this approach, the producer will keep inventory to avoid shortages: one for finished goods produced during the previous cycle and another for goods still in production during the current process.To summarize, the research omits both analyzing the integrated system's preference for various rework policies and incorporating IRI and IRS situations to determine the effects of the quality control rate on integrated and coordinated approaches.Because of this, no studies have examined the impact of verification and revamping procedures on throughput to ensure the viability of derived solutions while considering defective items.

Assumptions
1.A single manufacturer and retailer comprise the incorporated inventory control system.2. The demand rate is constant and known over the planning horizon.
3. The asynchronous rework policy and the rework procedure are at work here.4. As per Jaber and Khan [18], the suggested equal-sized distribution policy, the retailer's purchase, and the manufacturing policies of the manufacturer are as follows: given the proportion of faulty items, the manufacturer continuously produces Q units at P r . 5. The processing and supervision processes are carried out concurrently.Also, two views can be derived from the relationship between the manufacturer's safety check rate and production rate: The retailer orders q i (i.e., DT i ) Q units during the order cycle with perfect quality.Then, new units of the highest possible quality are checked and shipped to the merchant once each cycle until the manufacturer's inventory level is zero.
6.A lot is accepted only if the sample contains no defective items and no missorted items due to a Type 1 error, denoted by E 1 , or if the model includes more than one defective item and misclassification due to a Type 2 error, marked by E 2 .7. The producer verification rate is lower than its consumption rate, according to Bazan et al. [14].

Model formulation
The retailer's and the manufacturer's order policies can be expressed as follows, using the approach to equal-sized supplies that Jaber and Khan [18] suggested.The retailer orders q i (i.e., DT i ) units of flawless quality during the order cycle T i according to market requirements D. Given the number of defective goods, the producer consistently turns out units at a rate of R p .Then, until the manufacturer's inventory is depleted, new units with the highest possible quality are examined and sent to the retailer once every cycle.

Classification of acceptance probability P a
The basic definition of sampling is the probability that the lot will be accepted and, therefore, shipped to the retailer is 2 1 ( 1) ( 0) (1 ).
The quantity of flawed items in the sample of n is the random variable X in this formulation.Therefore, a lot is only accepted if it contains no defective items and no misclassification due to a Type 1 error, denoted by E 1 , or if it contains more than one defective item and there is a misclassification due to a Type 2 error, denoted by E 2 .We consider E 1 and E 2 constants and aspects of the sampling procedure.The likelihood that the lot won't be approved and sent for screening is 1 − P a .The random variable X is the number of defective items in an n-item sample.If X is approximately a binomial random variable with a probability distribution function, if the lot size is assumed to be large If the acceptance number X is set to zero, the likelihood that the sample contains no defective items is Therefore, the likelihood the lot will be accepted as described in (1 (1 ) ) (1 ) (1 ) ( . 1 ) ( 1)

Rework priority policy under IRI scenario
Rework-priority policies require the start of the reworking procedure by the manufacturer for damaged goods before sending finished goods to the retailer.In Figure 1, we consider a firm with a flawed production system.Manufactured goods are divided into two categories: defective and non-defective.As products are produced in batches or lots, the manufacturer will still have a substantial inventory of manufactured goods.It is presumed that the acceptance number is zero.Although lots can be misclassified with frequent Type 1 and Type 2 errors, the inspection procedure is not flawless.The anticipated lot size will be sent to the perfect goods warehouse if P a is the likelihood that a lot is accepted.The manufacturing rate of the enterprise, in this case, is more significant than its checking rate (α < P r ), which causes the production cycle of the enterprise to be shorter than its inspection period, i.e., 1 r L L T < and consider Q = m i DT i (i = 1 IRI scenario, i = 2 IRS scenario).After the entire inspection procedure is finished, defective products are reworked in these cases.Since damaged goods can be repaired and distributed as usual.
1.The producer creates Q total bunches of units (i.e., m 1 q 1 ) with a production rate of P r during the production process L p (note the dotted line in Figure 1).2. As part of the inspection procedure L i , the manufacturer's perfect products are gathered at a rate of α(1 − µ), which includes q 1 (1 − µ) units of perfect items.3.For the remainder of the first cycle (L p − L i ), the producer begins to fix partially flawed products into flawless ones at a rapid rate of R w , guaranteeing the manufacturer's inventory for perfect items equals the retailer's first lot q 1 .Every T 1 cycle, the manufacturer completes the reworking process and sends q 1 units to the store in the highest possible quality.Meanwhile, an asynchronous rework policy is applied to defective items.The manufacturer will charge for returned items.Depending on their quality, returned items are sent to the rework stage or to salvage.The lot is sent to the screening stage, where the products are separated into non-defective and reworkable if the inspector deems the lot unacceptable.A percentage of lots of products can be fixed, and these products can only be sold in the market.Reworkable items can be fixed at R w per time unit using the rework procedure.The manufacturer thus maintains three different types of inventory: work-in-process, inventory of non-defective goods sent to the main market, and inventory of reworkable goods.
A manufacturer who produces large lot size needs to send their product to the primary market by acceptance in cycle T 1 .
The retailer's first order quantity is The length of the production is The length of the reworked item, which includes misclassification goods, is .
The length of the inspection time is Since total production amount for the perfect item during 1

L
T is equal to retailer order quantity, Reworked for the defective items 1 1 .
A flawless production process leads to flawless production quantity is Under the rework priority policy, to fulfill the retailer's initial order lot, it is also necessary to rework partially damaged items and produce and inspect q 1 units.The following statement can be summarized by noting that the manufacturer's number of shipments should not be fewer than 1, 1, ABCD manufacturer's inventory with defective items is .
The manufacturer's accumulated inventory of perfect items is (1 ( 1) ) ( 1) Setup cost per production The manufacturer's total production of goods for flawless items during the manufacturing period (1 ) .2( 1) The holding cost for defective items incurred by the producer is Expected sales revenue is ( ) The total cost included the manufacturer's setup cost, production cost, inventory holding cost for defective items and perfect items, and transportation cost.The total cost per unit of time in RPP is 1) The expected net total profit is = Sales revenue -Total cost ( ) (1 ) ( 1) 1 (1 ) 2( 1) where 1 2 2 .( 1)( 1)

Discussion of the IRS scenario
In this part, the manufacturer's inspection rate exceeds its production rate, i.e., (α ≥ P r ).Under these conditions, the inspection process can be concluded when the entire production run is finished, indicating that Later, the remanufacturing process can start at the end of the entire manufacturing process.Moreover, the variation of inventory level under the RPP is shown in Figure 3.
Manufacture's inventory of perfect items Production time

Rework priority policy under IRS scenario
Under the IRS of RPP, the manufacturer's production amount includes manufacturing and safety checks, and (1 ) ( ), which can be rewritten as (1 ) 1 .
As a result, in both the IRI and IRS scenarios, the probability of failure always limits the manufacturer's number of RPP consignments.Furthermore, it is clear that 2 2 .
In addition, under the IRS method, Figure 4 shows that the manufacturer production rate for perfect items is Q(1 − µ).According to the graph, ABDC includes ABC and BCD, indicating the manufacturer's level of inventory was compiled with defective items per manufacturing ( ) Furthermore, perfect products from the manufacturer's stock level is The total cost function per unit time under RPP can be formulated as follows: Expected sales revenue is ( ) Expected total profit = Sales revenue -Total cost ( ) (1 ) 2 2 ( 1) where 1 2 2 .( 1)( 1)

Solution procedure
Our aim is to find the maximum profit.Theorem 1.The total profit function in equation TP 2 (22) is concave from an unknown variable Q. Proof.It is necessary to verify the profitability of the IRI scenario and to check its concavity in terms of the differential equation TP 1 (22) concerning Q ( ) ) We can find the Q value as ( ) Theorem 2. The total profit function in equation TP 2 (29) is concave from an unknown variable Q. Proof.Using a differential equation TP 2 (29) to test the IRS scenario to see if it can result in greater profitability concerning Q ( ) 2 .( 1)( 1) 2 .( 1)( 1) We can find the Q value as ( ) . ( 1) 1

Numerical examples
Take the example of a manufacturer who uses an imperfect manufacturing process.Before a lot can be dispatched to a store or market, it must undergo an approval sampling stage.Two methods are available to the manufacturer for assessing quality characteristics: destructive and non-destructive.A lot must not contain damaged goods to be sent to a retailer.In rejecting lots, they will proceed to a more costly screening stage, separating them into non-defective, reworkable categories.For two alternative rework policies, numerical examples are run under the IRI or IRS scenario to determine the best-integrated production inventory solution that considers defective goods.To demonstrate the feasibility of the operational solutions derived from the integrated system, the performance of the integrated system is compared.

Base case
The curves have been created for two values of E 1 and E 2 and three values of the probability of a defective item, µ = 0.1,0.2, and 0.3.Figure 5 shows the case when E 1 = 0.05 and E 2 = 0.04, whereas the sold lines are for that scenario.The sample sizes were chosen to range from 1 to 15. Table 2 shows that when the length of the cycle increases, the level of probability of acceptance will reduce.
After that defective rate, µ = 0.7 has the probability of acceptance being stable and maintained at 0.05.The same observation holds for sample sizes extending beyond the plot's right side and the plot's right side.The sampling scheme shows its maximum ability to detect the probability of error for sample sizes in the center of the range.From Table 2, µ = 0.1 and 0.2, the difference between the two defective rates at cycle length m = 5 is 0.2391, a significant difference.So, we conclude that P a is less responsive to µ for small and large n and more sensitive to µ for a medium sample size of n.    33) and get the Q value for the IRI equation (31); and the total profit is 73784230.54.Since α < P r , an increased upper limit of m 1 and an increase in µ eventually cause the integrated system to perform better under RPP.In particular, if µ is above a certain threshold, the probability of acceptance is 0.05.
Example 2. In the IRS scenario, i.e., α > P r , let us take values from the previous example except α = 350 per cycle and P r = 320, applying values in the equation (36) and we get Q = 1186.37,values together in the equation (34) that's the total profit is 196926073.36.In comparison of Examples 1 and 2, the IRS scenario gets 7.8% more profit than the IRI scenario at m 1 = m 2 = 15, i.e., same length of time.The defective increase is 0.5, and the cycle length is 15; the IRS scenario profit is higher than the IRI scenario profit.There is a 52.74% difference in profit between the IRS and other scenarios.As a result, IRI scenario drops its profit dramatically if defective rates increase in each scenario.

Sensitivity analysis
Through sensitivity analysis, acceptance sampling has significantly affected lot sizes, estimated net profit, and expected cycle length.Ideal lot sizes are positively affected by the probability of defective material but negatively predict net profit.
1. Selling price, manufacturing costs, inspection costs, manufacturing costs, and transportation costs do not change from the order of magnitude of the ideal.Still, there are differences in defective goods and holding costs for demand.2. Table 3 shows that in the IRI scenario, if demand decreases by 50%, profit will decrease by 23.74%, demand will decrease by 33.33%, and if demand increases by 50%, profit will increase by 57.82% and increase by 37.83%.3. Table 3 indicates the IRS scenario: if demand had decreased by 50%, profits would have decreased by 58.71%, whereas if demand had increased by 50%, profits would have increased by 92.97%.4. The transportation, inspection, production, and rework costs for both the IRI and IRS scenarios have no bearing on the overall profit or ideal order quantity.5. Rework is critical in all scenarios, with the IRI scenario decreasing profit by up to 67% and raising it by up to 65.62%.The IRS scenario improved profit by up to 93.62% and dropped it by up to 63.57%.6. Figure 6 demonstrates the variation in demand between the IRI and IRS situations.9.The analysis of the impact of Type 1 error probability concluded that it is more cost-effective to raise the lot size for higher values of mistake probability.10.It is discovered that a smaller lot size and a longer cycle time are required as the probability increases.
Additionally, the effects of the other model parameters have been explored.

Conclusion
This article presents the EPQ model for imperfect quality, the single acceptance model and misclassification errors, defective goods with an equal export policy, and manufacturer control over production, inspection, and rework processes.The IRI and IRS scenarios are the foundations upon which this study bases its analysis of the interaction between the manufacturer's total production and the inspection rate.Additionally, two separate rework policies are applied to the number of shipments in each situation.Optimal lot sizes that maximize objective functions are identified and demonstrated, and annualized net profit functions are also developed.The manufacturer's inspection and reworking processes may need to be fixed, which could significantly lower the impeccable production volume of the manufacturer's goods per unit of time and make it more challenging for the manufacturer to complete the product.The analysis of the impact of Type 1 error probability concluded that it is more expensive to raise the lot size for higher values of mistake probability.A study of the EPQ model is conducted to determine how it reacts to changes in the likelihood of Type 1 and Type 2 errors.It is discovered that, as the probability increases, a smaller lot size and a longer cycle time are required.The impacts of the other model parameters have also been studied and analyzed.

Further research
In additional research, the suggested paradigm does not permit the backorder.Trade credits and cost savings can help with this.Using the Joint Economic Lot-size Problem (JELP) model as an extension, this solution resolves the issues with traditional optimization strategies.It may be possible to divide the primary and secondary markets by reworking the priority problem and rewriting the delivery priority problem for multiple retailers.

1 hc
Holding cost for perfect items/unit /unit timeL pTotal production quantity per production run L r

2 LT 2 L
− L i is a partial reworking process.Because for perfect items, the retailer's first order lot equals the manufacturer's production volume amount during

Figure 5 .
Figure 5. Defective rate versus probability of acceptance

Figure 6 .
Figure 6.IRI scenario demand versus IRS scenario demand

Table 2 .
Defective rate and probability of acceptance Let us assume the values in the IRI scenario.Take cycle values is one-quarter period: a = 280 per unit,

Table 3 .
Parameter analysis