Proportion Of Defective Items Research Articles

Estimation of the lot fraction defective in a finite lot of products with auxiliary quality characteristics

Many companies use sampling plans for the acceptance or rejection of lots of products. The final outcome of this decision-making process is based on the inspection of a sample of products selected from a lot under inspection, where a quality characteristic is observed. An important parameter of interest related to acceptance sampling for attributes is the proportion of defective items. This parameter is unknown for a given lot of products, but it can be estimated from the aforementioned sample information. Additional quality characteristics can be observed at the inspection stage. We propose to use this auxiliary information to obtain more accurate estimators of the lot fraction defective at the estimation stage. Various relevant applications of this process are described. For possible scenarios that may arise in practice, the empirical properties of the suggested estimation methods are investigated using Monte Carlo simulations, and desirable results are obtained when there is a strong relationship between the quality characteristic of interest and the auxiliary quality characteristic.

A flexible planning approach for integrated lot sizing and rework planning with random proportion of defective products

We consider a stochastic capacitated lot sizing problem with in-line rework. Both defective and defect-free products can be the result of an imperfect production process in real-world production systems. However, the proportion of defective items in a production lot is subject to uncertainty. For economic and environmental reasons, the possibility to rework defective products appears reasonable since these products usually have considerable value. In this paper, a nonlinear model formulation for integrated lot sizing and rework planning is proposed. We use a sample average approach to approximate the generic nonlinear model formulation. To cope with the randomness of the proportion of defective products, we apply a flexible planning approach that allows the adjustment of production and rework quantities. We conduct extensive numerical investigations to evaluate the performance of the proposed planning approach.

Joint optimization of production lot-sizing and condition-based maintenance in an imperfect production process with dependent indicators

ABSTRACT This paper addresses the integrated optimization of the economic manufacturing quantity and the condition-based maintenance policy of a deteriorating manufacturing system. The considered facility produces a single type of product and is inspected at the end of each production run. Upon inspection, two dependent indicators are revealed: one implies whether the production process is in control or not and the other one represents the deterioration level of the facility. The dependence between the two indicators is that whenever the facility deterioration exceeds a pre-determined level, the system becomes more vulnerable such that the production process may switch to the out-of-control state. In the out-of-control state, a proportion of defective items is fabricated. Considering the inter-dependencies between production process and machine degradation, this paper develops an integrated production and maintenance model to minimize the overall cost. The expected cost rate in the long run is taken as the objective function to assess the proposed model, where the joint optimization of the lot-sizing and the maintenance policy is developed. The problem is formulated in the context of a semi-Markov decision process and solved with the successive approximation method. A numerical example is given to illustrate the applicability of the proposed model.

A production-repairing inventory model considering demand and the proportion of defective items as rough intervals

This paper explores a production inventory model in an imperfect production system under a rough interval environment. The occurrences of shortages are allowed and are fully backlogged. In this paper, two models are presented. In the first model, the demand of the product is considered as a rough interval. In the second model, both the demand and the defective rate are considered as rough intervals. In reality, these two factors cannot be assessed exactly due to a lack of available data. Moreover, there are few scenarios where the demand of the product is assessed in two-layer information. The inner layer demand is obvious, and it may extend up to the outer layer depending on the situation. Also, the same thing may happen for the defective rate of the product. In those cases, it is more reasonable to express the demand and the defective rate as rough intervals instead of fuzzy or interval numbers. Based on the expected value of rough intervals, the necessary and sufficient conditions are determined analytically to obtain the optimal inventory policies. The main advantage of this method is that the optimal profit and the optimal lot size are determined as crisp numbers rather than rough intervals. Obtaining a precise value for these two quantities is desirable for any production manager. Numerical examples for both models are provided to illustrate the solution methodology. Sensitivity analysis of the optimal solutions concerning the key parameters is conducted for identifying several managerial implications.

Optimal production policy for economic production lot-sizing model with preventive maintenance

ABSTRACT In this study, we determine the optimal manufacturing quantity for an economic production lot-sizing model with a defective rate and preventive maintenance for a manufacturing machine. We investigate the properties of economic manufacturing quantity to minimize expected costs during the production run when a random defective rate for that product is considered using an in-control condition and an out-of-control condition. If a manufacturing machine produces a proportion of defective items which is larger than the lot tolerance percent defective, then a machine shifts from an in-control condition to an out-of-control condition, and it should have a maintenance service to decrease the possibility that it will produce imperfect-quality products. A research model is developed to determine optimal manufacturing quantity and optimal production time when a defective rate and maintenance service are considered. Numerical examples are given for the applicability of the methodology derived in the paper.

Empirical Bayes Prediction in a Sequential Sampling Plan Based on Loss Functions

The application of empirical Bayes for lot inspection in sequential sampling plans is usually conducted to estimate the proportion of defective items in the lot rather than for hypothesis testing of the variables’ process mean. In this paper, we propose the use of empirical Bayes in a sequential sampling plan variables’ process mean testing under a squared error loss function and precautionary loss function, for which the prediction is performed to estimate a sequence of the mean when the data are normally distributed in the case of a known mean and unknown variance. The proposed plans are compared with the sequential sampling plan. The proposed techniques yielded smaller average sample number (ASN) and provided higher probability of acceptance (Pa) than the sequential sampling plan.

Design of Fuzzy Sampling Plan Using the Birnbaum-Saunders Distribution

Acceptance sampling is one of the essential areas of quality control. In a conventional environment, probability theory is used to study acceptance sampling plans. In some situations, it is not possible to apply conventional techniques due to vagueness in the values emerging from the complexities of processor measurement methods. There are two types of acceptance sampling plans: attribute and variable. One of the important elements in attribute acceptance sampling is the proportion of defective items. In some situations, this proportion is not a precise value, but vague. In this case, it is suitable to apply flexible techniques to study the fuzzy proportion. Fuzzy set theory is used to investigate such concepts. It is observed there is no research available to apply Birnbaum-Saunders distribution in fuzzy acceptance sampling. In this article, it is assumed that the proportion of defective items is fuzzy and follows the Birnbaum-Saunders distribution. A single acceptance sampling plan, based on binomial distribution, is used to design the fuzzy operating characteristic (FOC) curve. Results are illustrated with examples. One real-life example is also presented in the article. The results show the behavior of curves with different combinations of parameters of Birnbaum-Saunders distribution. The novelty of this study is to use the probability distribution function of Birnbaum-Saunders distribution as a proportion of defective items and find the acceptance probability in a fuzzy environment. This is an application of Birnbaum-Saunders distribution in fuzzy acceptance sampling.

Optimal production run time for an imperfect production inventory system with rework, random breakdowns and inspection costs

In real-life manufacturing systems, the presence of defective products in a lot is inevitable. While these products may be just scrapped in the food industry, in high-tech industries where the final product is very expensive, they may be reworked at a cost. A common assumption in the literature is that the inspection time needed to identify defective items is completed when the production process ends. However, the assumption of continuous inspection during production complicates the analysis, making it impractical for most production systems, especially when the production rate is high, and the proportion of defective items is low, making continuous inspection during production very expensive. In addition, such factors as process deterioration or other uncontrollable factors in the production process may interrupt the production of the lot. To address these practical issues, this paper integrates inspection time and the failure of production facilities into an imperfect production inventory model with rework, where the production run time is a decision variable and an inspection process continues even after a production run; the paper demonstrates significant effects on the optimal solutions, with shortages not allowed. Under these assumptions, a mathematical model is derived, and the concavity of the expected total profit function is proved. Optimal policy is obtained by applying the analytic method. Special cases of the model are studied and a numerical example with sensitivity analysis is provided to draw insights. Moreover, this numerical example is used to compare general and special cases.

Importance of Quality Control Implementation in the Production Process of a Company

AbstractThe concept of quality has gained increasing importance over the last decades. Regarding the production process of a company, the quality control implementation allows companies to offer a higher quality product, which has positive influence on customer satisfaction. This paper aims to reflect the importance of statistical methods for quality control. Based on the two broad areas of statistical inference, parameter estimation and hypothesis testing, we demonstrate the usefulness of such methods in problem solving when the proportion of defective items is considered.

Importance of Quality Control Implementation in the Production Process of a Company

The concept of quality has gained increasing importance over the last decades. Regarding the production process of a company, the quality control implementation allows companies to offer a higher quality product, which has positive influence on customer satisfaction. This paper aims to reflect the importance of statistical methods for quality control. Based on the two broad areas of statistical inference, parameter estimation and hypothesis testing, we demonstrate the usefulness of such methods in problem solving when the proportion of defective items is considered.

Multiple Deferred State Sampling Plan with Fuzzy Parameter

Traditional multiple deferred state sampling plan by attribute is an inspection with crisp parameter. In this plan, in addition to using the current lot information, the future successive lots information can also be utilized on sentencing the current lot. In this study, a fuzzy multiple deferred state (FMDS) sampling plan by attribute is proposed when the proportion of defective items (p) is uncertain. So, the fuzzy probability theory is applied to construct the operating characteristic curve of mentioned plan. Then, we concentrate more on a special feature of FMDS(0,1,2) plan. We show that the proposed plan is well-defined because it converts to classic case when p is not imprecise. In order to be fully informed, some examples will also be discussed. The comparisons between the FMDS(0,1,2) plan and the existing fuzzy single sampling plan (FSSP) with zero and one acceptance number show that the proposed plan is more powerful than the FSSP in distinguishing good and bad quality of the lot. It also enjoys the least sum of the consumer and producer risks.

Optimal buffer inventory and inspection errors in an imperfect production system with preventive maintenance

This paper considers an imperfect production system with preventive maintenance to obtain the optimal buffer inventory and inspection policy for sold products with free minimal repair warranty. The production system is subject to a random movement from an in-control to an out-of-control state, where some proportion of defective items are produced by the production system during both the in-control and out-of-control states. Online inspection is continuing after a time variable during the production process. Another offline human-based inspection policy is considered at the end of the production cycle to identify the defective items. Defective items found by the inspector are salvaged at some fixed cost before being shipped and the non-inspected items are passed to the customer with free minimal repair warranty. During human-based inspection, some misclassifications may arise from the inspector’s side. Thus, two types of inspection errors (Type I and Type II) are considered to make the model more realistic rather than the existing models. A numerical example along with graphical representations are provided to illustrate the proposed model. Sensitivity analysis of the optimal solution with respect to major parameters of the system has been carried out, and the implications are discussed.

Optimal batch quantity in a cleaner multi-stage lean production system with random defective rate

Optimal batch size reduces variability in the lean manufacturing process with minimized system costs. It simplifies the scheduling, enhances the quality, reduces inventories, and improves the production process continuously. On account of this, determining optimal batch size is the area of interest for the researchers with the objective of reducing inventories and related system costs. Products of the textile sector are produced generally in a multi-stage production setup, and it consists of random defective rate within the long-run production systems. In this context, this paper revisits an economic production quantity (EPQ) model with an imperfect multi-stage production system and analyzes an inventory model by considering a random defective rate in a cleaner multi-stage lean manufacturing system. These defective items are reworked and converted into perfect quality items by incurring additional processing costs. The proportion of defective items can be reduced through continuous improvements in the production process reliability by performing various lean manufacturing techniques, enlisting the total productive maintenance at the highest rank. A mathematical model is developed using a familiar beta distribution density function for the random defective rate. Then, the total cost of the system is minimized through analytical technique, where the batch size is a decision variable. Outcomes through analytical and numerical study confirm that optimum batch size is obtained by this model and it has a direct relationship with number of production stages.

Optimal strategy for a manufacturer-retailer inventory system with defective items under retailer partial trade credit policy

In order to reduce default-risk, a retailer in practice frequently requests its credit risk customers to pay a part of the purchase amount as collateral deposit at the time of ordering, and then offers a trade credit for the outstanding balance (partial down-stream trade credit). In contrast, the retailer usually receives a full trade credit on the purchase amount from the manufacturer (up-stream full trade credit). The present study develops a single manufacturer single retailer integrated supply chain model under two-level trade credit in which the retailer receives a full trade credit from the manufacturer but offers a partial trade credit to its customers. The manufacturer delivers the retailer’s order quantity in a number of equal shipments. An arriving lot from the manufacturer contains a certain proportion of defective items. The market demand rate is sensitive to the length of the customer’s trade credit period which is offered by the retailer. The present study aims to determine the optimal replenishment cycle time, the order quantity and the number of lots from the manufacturer to the retailer such that the total profit of the integrated supply chain system is maximized. Solution algorithms are designed to find the optimal solution of the model. The theoretical results developed in the paper are illustrated through a numerical example. Sensitivity analysis of some important parameters is presented to study their influence on the optimal solution.

A fuzzy random EPQ model with fuzzy defective rates and fuzzy inspection errors

In this paper, we consider an economic production quantity (EPQ) model for imperfect production process under fuzzy random variable demand considering inspection errors. Due to the first stage inspection errors, some proportion of defective items are returned because of dissatisfaction of the custo mers. In the previous traditional models, the defective rates and the inspection errors follow some probability distributions. However, in real life situation, it is almost impossible to obtain the statistical information precisely. Thus, this study proposes the fuzzy defective rates and the fuzzy inspection errors. In addition, this model interpolates two more stages of inspections, one is after the production run time, and another is after the beginning of the rework process. The purpose of this study is to establish a fuzzy random EPQ model with the fuzzy defective rates and inspection errors. The expected profit per unit time is calculated by using fuzzy random renewal reward theorem. This model maximizes the expected profit per unit time in fuzzy sense. We develop a methodology for finding the global optimal solutions. A numerical example is also provided to illustrate our proposed model. Furthermore, sensitivity analysis is also carried out in order to present some managerial inferences.

Replenishment, production and quality control strategies in three-stage supply chain

In this paper, we propose to jointly integrate and coordinate production, replenishment and quality inspection decisions in a three-stage supply chain control problem. The transformation stage produces one final product type and responds to a stable market demand. After a random lead time, the supplier delivers raw materials in batches which may each contain a certain proportion of defective items. When a lot of raw materials is received, a lot-by-lot acceptance sampling plan is applied, and then a decision is taken with regards to a 100% screening or discarding of the sampled lot. In this article, we focus on the existing interaction between the applicable quality control decisions and the replenishment and production control decisions. The objective is to determine a control policy for production, replenishment and quality activities which minimizes the total cost, including purchasing costs, production and quality inspection costs, as well as the inventory/backlog costs. A simulation model and a Response Surface Methodology are used to find the optimal parameters of the proposed policy. The obtained results show that the integration of 100% screening or discarding decisions in a new “hybrid” one is more beneficial, and guarantees a better coordination at a lower cost.

Joint optimal lot sizing and production control policy in an unreliable and imperfect manufacturing system

This paper deals with the problem of the joint determination of the optimal lot sizing and optimal production control policy for an unreliable and imperfect manufacturing system, where the quality control of lots produced is performed using an acceptance sampling plan. The proportion of defective items, the time between failures and the time to repair are generally distributed. The incurred total cost includes manufacturing cost, transportation cost, inspection costs, rejection cost of defective items, replacement cost for returned defective items from customers, and holding and backlog costs. The associated cost minimization problem is formulated with a stochastic dynamic programming model where the lot sizing and production rate are considered as decision variables. Given the difficulties in solving such a highly stochastic model analytically or numerically, we adopted a modified hedging point policy (HPP) to control the production rate, as well as an economic lot sizing policy for batch processing control; we also relied on a simulation-based experimental approach to determine a close approximation of the optimal control parameters. It is shown that production should be accelerated at the maximum production rate, not only when building the safety stock, as in the classical HPP, but also after rejecting a lot, in order to recuperate the loss in inventory and to maintain the on-hand safety stock. Numerical experiments and thorough sensitivity analyses are provided to illustrate the effectiveness of the proposed control policy and the robustness of the resolution approach. Some interesting behaviours regarding the impact of different parameters on the optimal decision variables are observed and discussed.

A multi-stage production-inventory model with learning and forgetting effects, rework and scrap

This paper studies a serial production line where a proportion of defective items is produced at each stage. Defective units enter a rework process, which is imperfect as well. Twice defective items are scrapped. This paper also considers learning and forgetting in production and rework processes and studies how the number of shipments of a lot from a production stage to the next influences the overall performance of the system. A model for a multi-stage production-inventory system is developed and optimized against an aggregate performance measure of four partial measures that are based on production time, process yield, in-process inventory and shipment frequency. Each of these partial performance measures is weighed by the system’s decision maker in accordance to importance. The numerical results show how the values of learning rates, weights assigned to the partial performance measures and the number of production stages influence the overall performance of the system.

An integrated vendor–buyer cooperative inventory model in an imperfect production process with shortage backordering

In this paper, we develop an integrated inventory model for vendor–buyer coordination under an imperfect production process. The proportion of defective items in each production lot is assumed to be stochastic and follows a known probability density function. The vendor inspects the items while they are being produced and delivers good-quality items to the buyer in small lots over multiple shipments. We assume that shortages are permitted and are completely backordered. The expected annual integrated total cost is derived and a solution procedure is provided to find the optimal solution. Numerical examples show that the integrated model gives an impressive cost reduction in comparison to an independent decision by the buyer. The results also show that if back orders are permitted, the expected integrated annual cost is less than that when back orders are not permitted. Therefore, if customers are willing to wait for the next shipment when a shortage occurs, it is profitable for the company to allow back orders although the delay incurs a penalty cost.

Using simple methods to derive EOQ and EPQ models with shortage and imperfect quality

Huang [ Journal of Statistics and Management Systems, Vol. 6 (2003), No. 2, pp. 171– 180 ] studied the EOQ (Economic Order Quantity) and EPQ (Economic Production Quantity) models with backlogging and defective items using the algebraic approach. He assumed 100% inspection policy and the known proportion of defective items was removed prior to storage or use after the screening process. In this paper, we will off er both basic inequalities, arithmetic-geometric mean inequality (AGM) and Cauchy-Bunyakovsky-Schwarz inequality (CBS), to derive both the optimal lot size and backorder level under the total relevant cost per unit time minimized. Without the method of completing perfect square, the proposed methods derive the global minimum of the total relevant cost per unit time more easily than the algebraic approach used by Huang (2003)