The economic production quantity model with optimal single sampling inspection

Abstract

1. Introduction

The Economic Production Quantity (EPQ) model is a well-known and widely used technique in production management. Under the traditional formulation, the quality of produced items is assumed perfect, and non-conforming (NC) items are not present. However, in real-world applications, NC items may be produced and that should influence the decision of the EPQ. Therefore, in a real-world planning, a policy for screening of products in addition to determining of order quantity is necessity. To consider the NC items, this study develops the EPQ model with a single optimal sampling inspection. The model minimizes the expected total cost, including inventory and quality costs, while fulfilling the quality and risk requirements of a producer and consumers.

In this paper, the economic production order quantity, sample size and acceptance number are determined in a manufacturing system that products are ordered in batches or lots. The lot is inspected by taking a single random sample. There are various sampling plans in the literature. The single-sampling plan is the most common acceptance sampling plan (Veerakumari & Harikrishnan, 2019) that is a simple plan to use. In this sampling, one random sample is selected. The lot is accepted if the number of defects in the inspected sample is equal or less than a, where a is the acceptance number (Nakhaeinejad, 2019). The lot is rejected if the number of NC items exceeds the acceptance number. A 100% inspection should be done for rejected lots. The NC items revealed by the inspection are thrown away and not replaced by perfect items.

Although there are some researches that determine the optimal batch production of the defective production system, the policy of single sampling inspection in the EPQ model by considering both EPQ- and quality-related costs has not been explored. The new proposed model obtained the economic production order quantity, sample size and acceptance number simultaneously while considering the producer’s and the consumer’s quality and risk. In the following, a brief discussion of related researches is given.

Yoo et al. (2009) proposed the EPQ model incorporating both imperfect production quality and two-way imperfect inspection issues. They considered Type I and Type II inspection errors. They also investigated rework and salvage for screened and returned items. Tu et al. (2011) determined the optimal lot size and the backorder level by minimizing the total relevant cost. They showed that their method derives the global minimum more appropriately than the literature. Pan et al. (2012) studied EPQ model with statistical process control and maintenance. They adopted control chart with Taguchi’s loss function to estimate the quality cost. They determined the optimal control chart parameters and planned a corresponding maintenance policy. Hsu & Hsu (2013) proposed two EPQ models. They considered imperfect production processes, inspection errors, backorders and sales returns in their model. They obtained the optimal production lot size and the maximum shortage level. Roy et al. (2014) deals with an economic production lot size model with defective items considering stochastic demand, backlogging and rework. They considered exponentially distribution for shifting the process from ‘in-control’ state to an ‘out-of-control’ state. Hsu & Hsu (2014) studied EPQ models with imperfect production processes. They determined closed-form solutions to decide the optimal production lot size and backorder quantity. Qin et al. (2015) explored a zero-defect single-sampling for assembly lines. They determined inspect sample size for each part, when the resources are constrained and a NC risk of product is not a linear combination of individual parts NC risks. Al-Salamah (2016) developed an EPQ model with the imperfect production process and inspection. He determined the optimal lot size for batch production where the batches before sent out to the market are inspected by a destructive or non-destructive sampling. Das et al. (2017) proposed a model for NC items in integrated production and inventory systems. In their model, the manufacturer sells products to the buyer in some lots, and then the buyer sells the products to the customer after inspection. They considered the first type of Beta distribution for the defective item ratio and the Normal distribution for the lead-time demand. Cheikhrouhou et al. (2018) considered an inventory model with a lot inspection policy. They studied two cases: (i) the defective lots send back to supplier with retailer’s payment, and (ii) the retailer sends defective products while receiving next lot from supplier with supplier’s investment. Their results suggest the first case is better than the second one. Cheng et al. (2018) studied an optimal vendor–buyer inventory model with defective items. They considered that the vendor manages the inspection process. They proved that the annual cost function is convex. They obtained the order quantity, the number of shipments and the number of defective item disposals. Nakhaeinejad (2019) proposed a two-step methodology for conducting a single-sampling inspection. For generating different single sampling inspection plans, he runs an optimization model in the first step. Then in the second step, he ranked the inspection plans by the combined methodology of Shannon Entropy Approach and Linear Assignment Method. Veerakumari & Harikrishnan (2019) developed economic order quantity (EOQ) model to minimize the sum of risks and considering the trade credit possibility. They proposed the optimal design by minimizing total costs, including the costs of inspection, stock holding and ordering. Cheng Guo et al. (2018) considered reliability and quality degradations by integrating the lot sizing, quality control and condition-based maintenance. They used 100% inspection policy to obtain the defectives proportion. When the proportion reaches a given threshold, an overhaul action is conducted. Al-Salamah (2019) presented inventory models in an imperfect manufacturing process. He proposed two rework strategies that can be asynchronous and synchronous with rework rate, which is different from the production rate. He proved that the type of strategy used for rework and the rate of rework influence the lot size and backorder, especially for defective proportion large values. Öztürk (2020) considered defective items in the various lots of an ordered shipment and chooses a sub-lot inspection policy. He developed mathematical models for two cases for defective items. He showed a close relation between the optimal order size and sample size, which can maximize the total profit. Khalilpourazari et al. (2020) studied a robust multi-item EPQ model for defective items with allowable rework. Their model minimizes the total inventory costs and total required warehouse space. They used Multi-Objective Grey Wolf Optimizer and Multi-Objective Water Cycle Algorithm for solving their proposed model. Nakhaeinejad (2021) proposed a zero-defect single sampling for inventory model that the received orders may contain defective items. He showed the effectiveness of the proposed model by comparing with the traditional EOQ model. Öztürk (2021) studied the issue of production facilities failure in inspection processes in which rework and shortages are not allowed. He considered production run time as a decision variable. He demonstrated significant effects on the optimal solutions and obtained optimal policy by applying the analytic method. Thomas & Kumar (2022) developed a fuzzy EOQ model by considering inspection errors in single sampling plans. They considered the rate of an order turn to be scrap, the costs of holding and the back-orders by fuzzy random variables and obtained maximum total expected profit. Hauck et al. (2022) considered EPQ model with two stages of quality control, including early stage screening after the most critical operation and final screening. They considered the impact of the screening time on both screening costs and the defect detection rate to determine the optimal lot size jointly with the optimal screening time. Their results show the importance of careful estimation of the defect rate. Heydari (2022) studied a problem that a manufacturer buys a second-hand product from one customer and sells it with a two-dimensional warranty to another customer. The inspection model examines the product during the warranty period to identify and eliminate hidden defects. In his model, the number and also the time of inspections are determined to minimize the service cost during the warranty period under periodic and sequential inspection policies. Table 1 outlines the key characteristics of this study in comparison to the literature of the EPQ models with defective items.

The literature review shows there is less attention in addressing EPQ model with NC items. However, in real-world production systems, NC items may be made, whereas in a classical EPQ model, defective items are not presented. Although some studies have explored the EPQ model with imperfect quality, they have not investigated the joint minimization of EPQ and quality costs. Conversely, this paper contributes the joint EPQ inspection model that defines a single sampling inspection policy in the EPQ model to consider defective items. The proposed model considers both quality and EPQ-related costs. In addition, the model takes quality and risk requirements of producer and consumer. In the EPQ aspect, ordering cost and holding cost are calculated, whereas in the quality control subject, the costs of inspection and NC items take into account to determine the production order quantity as well as the sample size, and acceptance number. Moreover, in this study, the probability distribution for NC items in the lot is assumed to be the Binomial distribution and the probability distribution for the number of NC items detected by inspection is supposed to be a Hypergeometric distribution.

The remainder of this paper is as follows. Problem description and notations are presented in the next section. Section 3 describes the developed model for the EPQ model with inspection. The solution approach for the proposed model is explained in Section 4. A numerical example is presented and discussed in Section 5. In Section 6, numerous examples are generated for sensitivity analysis and comparison study. Finally, Section 7 concludes the paper and presents future research directions.

2. Problem description and notation

In this paper, determining EPQ in process with imperfect production is considered. The necessity of considering imperfect production in determining EPQ is related to real-world production, where NC items may also be produced. In this regard, the produced products should be checked whether it is perfect or imperfect. The probability of producing an NC item is r. The production rate is finite with P units per time. The lot size is Q. As Fig. 1 shows, an order is shipped to one process of production. As defective items may be produced, a sample with size n is taken from the lot before doing the following production process. Through the single-sampling inspection, acceptance or rejection of the lot is examined. The lot is accepted, if the number of defective items in the sample is equal to or less than ‘the acceptance number, a’. In the accepted lot, the NC items, which are revealed through the inspection process, are thrown away and not replaced by perfect items. If the number of defective items in the sample is more than ‘a’, the lot is rejected for 100% screening and the NC items are thrown away. The proposed single sampling inspection plan considered both the maximum level of risk for the producer (AQL, the probability of accepting a defective lot) and the maximum level of risk for consumers (LTPD, the probability of rejecting a perfect lot).

Fig. 1.

Production route in the proposed model.

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In the problem under investigation, the total cost, including EPQ and quality costs, is minimized fulfilling the producer’s and consumers’ requirements regarding quality and risk. In addition, the optimal production lot, Q, the sample size, n, and the acceptance number, a, are determined.

The following notations are used throughout this study:

Variables:

Q: the lot size that is ordered.

n: the sample size for inspection.

a: the maximum number of NC items in the lot to be accepted (acceptance number).

Parameters:

D: the number of units demanded per time unit.

P: the number of units produced per time unit.

d: the number of NC items in the lot.

r: the rate of NC items in the lot.

b (d| Q, r): the probability of existing d defective items when the order size is Q and the NC rate is r.

h (a| Q, d, n): the probability that the number of defective items is equal to or less than ‘a’ in the inspection process when the acceptance number is a, the order size is Q, the number of NC items is d and the sample size is n.

ERQ₁: the expected reduction of the order quantity when the lot is accepted.

ERQ₂: the expected reduction of the order quantity when the lot is rejected.

TERQ: the total expected reduction of the order quantity.

A: the fixed ordering cost.

h: the unit holding cost per time unit.

c: the unit NC cost of one defective item.

l: the inspector labour cost per time unit.

t: the inspection time of one item.

AQL: the average quality level.

LTPD: the lot tolerance percent defective.

|$\alpha$|⁠: type I error or producer risk.

|$\beta$|⁠: type II error or consumer risk.

EOC: the expected ordering cost.

EHC: the expected holding cost.

ENC: the expected NC cost.

EIC: the expected inspection cost.

ETC: the expected total cost.

3. Model formulation

In the EPQ model, the objective function is considered total cost minimization. The total cost includes ordering and holding costs. The total objective function, Equation (1), included ordering cost, Equation (2), and holding cost, Equation (3).

By minimizing the total cost function in Equation (1), the production order quantity (EPQ) is obtained as follows.

In the minimum cost obtained for the EPQ model, it is assumed that all units produced are perfect. However, in practice, defective products may also be made. This paper proposes an inspection policy for the EPQ model to consider defective items. The inspection policy that considered in this paper is single sampling according to the fact that it can screen the products in a simple and efficient way. The defective items discovered during the inspection are assumed to be thrown away and not replaced by good items.

In the single sampling, two cases may occur. In Case 1, the number of defective items is at most ‘a’ in the sample. To be specific, a denotes the acceptance number of defective items. Therefore, the lot with k reduction in the production order quantity is accepted, and this accepted lot is placed in the next production stage. As a remark, k is the number of NC items, which are revealed through the sampling. Case 2 occurs if the number of defective items in the inspected sample is greater than a. In such a case, the lot is rejected for 100% inspection, and the production order quantity is reduced due to discarded defective items.

Case 1. The lot is accepted.

When the lot is accepted, defective items in the inspected sample are equal to or less than the acceptance number, a. Under Case 1, ERQ₁, the expected reduction of production order quantity is modelled in Equation (5).

where

In this case, there is the reduction in production order quantity due to defective items found in the selected sample, and there is the cost of NC items due to non-inspected portion of the lot.

Case 2. The lot is rejected.

When the lot is rejected, the reduction in production order quantity is due to the defective items found in the lot because it has to go through 100% inspection. In this case, the cost related to NC items did not occur due to 100% inspection. The expected reduction in production order quantity is calculated as follows:

where

The probability of existing defective items in the lot follows a Binomial distribution, which is expressed in Equation (6). In addition, the number of defective items in the sample found by inspection follows a hypergeometric distribution, which is presented in Equation (8). Now, the total expected reduction in the production order quantity, TERQ, can be obtained according to the expected reduction in the two possible cases as follows:

So, the production order quantity is calculated as follows:

Based on the production order quantity calculated in Equation (10), the proposed inspection model minimizes the expected total cost. The expected total cost includes the (i) expected ordering cost, EOC, (ii) expected holding cost, EHC, (iii) expected NC cost, ENC and (iv) expected inspection cost, EIC.

Equations (11–14) express EOC, EHC, ENC and EIC per cycle.

Therefore, the expected total cost, ETC, is modelled as follows:

Equation (15) minimizes the ETC, whereas the producer’s and consumer’s level of quality and risk requirements are considered to specify the optimal production order quantity (Q), the sample size (n) and the acceptance number (a). Following the study of Veerakumari & Harikrishnan (2019), the sum of the producer’s and consumer’s risk can be presented as follows:

where |${P}_a(AQL)$| and |${P}_a(LTPD)$| are the probabilities of accepting the lot when the quality level is AQL and LTPD, respectively. As a remark, AQL is the maximum of percent NC items that is acceptable based on the producer’s product quality, and LTPD is the high level of NC items that is unacceptable for consumers. From the producer’s point of view, a lot with the maximum defective level of AQL should have a high probability of accepting, and from the consumer’s point of view, a lot with a defect level as high as LTPD should have a low probability of acceptance. The producer’s risk defines as the probability of rejecting a lot at the quality level AQL, whereas the consumer’s risk defines as the probability of accepting a lot at the quality level of LTPD. The risks of producer and consumer are denoted by α and β, respectively (Subramani & Balamurali, 2016).

The mathematical model for the proposed problem is expressed in Equation (17), which includes quality costs and EPQ costs as well as take the of producer’s and consumer’s risk.

4. Solution procedure

To solve the problem, the proposed model considered as an integer model, since all decision variables (production order quantity, Q, sample size, n, and acceptance number, a) take only integer values. So, the optimization methodology used for minimizing Equation (15) could be direct search due to Q, n and a as decision variables are discrete and have the upper bound. The proposed solution procedure is shown in Fig. 2.

Fig. 2.

The optimal solution procedure for the proposed EPQ model with the inspection policy.

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As shown in Fig. 2, the objective function in Equation (15) is calculated for different possible values of n (⁠|$1\le n\le Q$|⁠) and production order quantity (⁠|$1\le Q\le D$|⁠). In addition, the acceptance number, a, is determined through Equation (18). Specifically, Equation (18) minimizes the sum of the producer’s and consumer’s risk in single sampling plan with the sample size n such that constraints, which are modelled in Equation (17), are satisfied (Veerakumari & Harikrishnan, 2019).

5. Numerical example

The proposed model of this paper is helpful for producers who deal with issues, including EPQ, inspection policy, sample size and acceptance number. In addition, the proposed EPQ model with an inspection policy is useful especially for batch production companies. Consider a production company with a similar production process to Fig. 1. The manufacturing process is not perfect, so before lots can be shipped to the next production process, the lots go through inspection operations. In inspection operations, the lots may be accepted or rejected. The accepted lots be sent out directly to the following production process, whereas the rejected lots are sent to screening stage before shipping to the next production process. Suppose that the manufacturer has the parameters listed in Table 2. These parameters show the base case of the numerical examples in this paper. The proposed solution procedure in section 4 is applied to obtain optimal solution.

The proposed procedure has been coded in MATLAB software. The results of running the developed EPQ model with the inspection for parameters in Table 2 are Q = 198, n = 68 and a = 2, with the minimum total cost ETC = 1074.929 where the cost components are EOC = 161.8254, EHC = 216.2826, ENC = 181.3318 and EIC = 515.4893. Note that without inspection policy, the optimal decision for EPQ is Q = 160.3567 according to Equation 4. In this case, the expected total cost is 2264.1657 with the cost component EOC = 187.0829, EHC = 187.0829, ENC = 1890 and EIC = 0.

Comparing the results with the two EPQ models demonstrates that ‘the proposed EPQ model with inspection’ considerably decreases the expected total cost. This finding highlights the effectiveness of the proposed EPQ model in managing production quantity while minimizing the total cost.

6. Sensitivity analysis and comparison

This section proposes numerical sensitivity analyses on key parameters (A, h, r, c, l and t). When the target parameters are varied, the other parameters are kept unchanged. In the sensitivity analysis of each parameter, the performance of the ‘proposed EPQ model with inspection’ compares with the ‘classical EPQ model’. The classical EPQ model with the objective function in Equation (1) and the cost components in Equations (2) and (3) is calculated for numerical examples of Tables 3–8, and Equation (4) is used to calculate the optimal order quantity of the EPQ model without inspection. For the proposed EPQ model with inspection, Equation (17), is solved based on the proposed procedure displayed in Fig. 2. The results of solving the proposed model for numerical examples in Tables 3–8 show the efficiency and validity of the proposed model in comparison with the classical EPQ model. The results in Tables 3–8 are shown in Figs 3–8.