Multidimensional Mixed Sampling Plans

Mixed sampling plans are two-stage sampling plans in which variable and attribute quality characteristics are used in deciding the acceptance or rejection of the lot. Due to modern quality control systems, mixed sampling plans are widely applied in various stages of production. Use of different sampling plans for different quality characteristics would result in loss in economy, time, and labor. Therefore an attempt has been made to design multidimensional mixed sampling plans (MDMSP). Based on multidimensional quality characteristics, a MDMSP aims at controlling overall quality of a lot or process. The design aspect of a MDMSP is given in detail based on the Poisson model (type B process) in the second stage. Tables and illustration are also provided. Suresh and Devaarul (2000) have designed mixed sampling plans with chain sampling as the attribute plan. Suresh and Devaarul (2003) have developed mixed sampling plans for maximum allowable variance. Suresh and Devaarul (2003) have combined process control and product control to reduce sampling costs. Schilling (1967) has given a method for determining the operating characteristics of a mixed sampling plan including several other measures of the plan. A multidimensional mixed sampling scheme consists of two stages in which several variable and attribute quality characteristics are considered in deciding the acceptance or rejection of the lot. The main advantage of a MDMSP over any other plan is the reduction in the sample size for the same amount of protection.     

Developing a variables multiple dependent state sampling plan with simultaneous consideration of process yield and quality loss

Acceptance sampling plans have been utilised predominantly for the inspection of outgoing and incoming lots; these plans provide effective rules to vendors and buyers for making decisions on product acceptance or rejection. Multiple dependent state (MDS) sampling plans have been developed for lot sentencing and are shown to be more efficient than traditional single sampling plans. The decision criteria of MDS sampling plans are based on sample information not only from the current lot but also from preceding lots. In this study, we develop a variables MDS sampling plan for lot sentencing based on the advanced process capability index, which was developed by combining the merits of the yield-based index and loss-based index. The operating characteristic function of the developed plan is derived based on the exact sampling distribution. The determination of plan parameters is formulated as an optimisation model with non-linear constraints, where the objective is to minimise the sample size required for inspection and the constraints are set by the vendor and the buyer to satisfy the desired quality levels and allowable risks. The performance of the developed plan is examined and compared with traditional sampling plans. A step-by-step procedure is provided, and the parameters of the plan under various conditions are tabulated for practical applications.     

Proposal of variable sequential sampling plan having desired operating characteristics indexed by quality loss

The proportion of non-conforming items has been traditionally utilised as an evaluation criterion for quality of items. However, the proportion of non-conforming items is not necessarily useful as a proper evaluation criterion for controlling high-quality manufacturing in recent years. Accordingly, in order to achieve further quality improvement and innovation, more careful quality evaluation has been required newly. Then, a concept of quality loss in the Taguchi methods has been devised as a severe criterion of quality evaluation. Hereby, a variable single sampling plan having desired operating characteristics (OCs) indexed by quality loss has been proposed in the area of statistical quality control. By the way, the most economical sampling inspection in the average sample number (ASN) is the sequential sampling plan based on the Wald’s sequential probability ratio test. Then, from the viewpoint of cost reduction, we discuss a variable sequential sampling plan having desired OC indexed by quality loss with the aim of expansion of the utility of variable sampling plan for quality loss. As the result, the design procedure of the sequential sampling plan for satisfying some required design conditions indexed by quality loss is provided. In addition, the effectiveness of the proposed sequential sampling plan is verified through some numerical examples.     

Dependent Mixed and Mixed Repetitive Sampling Plans for Linear Profiles

In this paper, we present two new dependent mixed sampling plans and a mixed repetitive sampling plan based on the process yield index for linear profiles. A mixed sampling plan includes two stages; the first stage is a combined test by variables and by attributes, and the second stage is tested by attributes only. The first plan is based on the traditional mixed sampling scheme without marginal quality, and the second plan is based on the modified mixed sampling scheme with marginal quality. If a lot is very bad, the probability it can be rejected with the small initial sample is higher for dependent mixed sampling with marginal quality. We compare dependent mixed plans with other types of double sampling plans; the former outperforms the others with respect to average sample number. Additionally, the number of profiles required in the first stage is much smaller than in the second stage. When the inspection is costly and destructive, a mixed repetitive sampling plan includes one stage that combines the inspection by attributes and by variables and allows resampling in some conditions. We compare the mixed repetitive sampling plan with the repetitive sampling plans by attributes and by variables. The probability of acceptance using the mixed repetitive sampling plan performs better than the repetitive sampling plans by attributes and by variables. Three examples are used to illustrate the proposed methods. Copyright © 2017 John Wiley & Sons, Ltd.     

Minimum Average Total Inspection Plans for Variables Sampling, Indexed by Average Outgoing Quality Limit

Dodge and Romig's 1944 volume Sampling Inspection Tables provided, among other procedures, sampling inspection plans for attributes inspection which were indexed by average outgoing quality limit. These tabulated plans guaranteed that when rejected lots are appropriately treated, the long-term average outgoing product quality would not exceed the average outgoing quality limit (AOQL) index of the plan. In addition, the plans were structured so that when the submitted quality level was at the indicated level of process average, the total amount of inspection would be minimized. In this article, plans are presented that match the AOQL, lot size, and process average parameters of the Dodge and Romig plans, but are based on variables sampling rather than attributes sampling. It is assumed that the distribution of the measured characteristic is Normal. Although measurements data are usually more difficult to obtain than attributes data and more calculation is involved in using the plans, the sample sizes are generally substantially smaller. The exception to this is for unknown standard deviation plans, higher AOQL index, and process average near the listed AOQL. It would be presumed that in present-day applications, plans involving low levels of both process average and AOQL would be most widely used.     

Chain Sampling Scheme under Constant Inspection Errors

The primary aim of this paper is to extend the inspection error consideration to chain sampling schemes, an area that has not been dealt with in the literature. A mathematical model is developed to investigate the performance of chain sampling schemes under constant inspection errors. Expressions of performance measures, such as operating characteristic function, average total inspection and average outgoing quality, are derived to aid the analysis of a general chain sampling scheme, ChSP‐4A ( ) r, developed by Frishman. This study reveals that as Type I inspection error increases the probability of acceptance will decrease and as Type II inspection error increases the acceptance probability will increase. The effect of Type II error on the probability of acceptance is very marginal compared with that of Type I error, especially when the true fraction non‐conforming is small. In addition, the effects of inspection errors can be ‘eliminated’ by transforming to its equivalent perfect inspection counterpart, hence greatly reducing the complexity of the analysis. The effects of other sampling parameters are also studied to serve as a foundation for future plan designing purposes. Copyright © 2006 John Wiley & Sons, Ltd.     

A variables multiple dependent state sampling plan based on a one-sided capability index

Acceptance sampling plan has been considered as one of most practical tools for quality assurance applications. While various types of acceptance sampling plans have been developed for different purposes, single acceptance sampling plan is the most popular because it is simple to administrate. However, a new concept called multiple dependent state sampling has gained the attention of scholars in recent years. The underlying principle is that the acceptance of a submitted lot should not only depend on the quality of the current lot but also consider the quality of the preceding lots. This research develops a variables multiple dependent state sampling plan (VMDSSP) for unilateral specification limit based on a one-sided capability index. The operating characteristic (OC) curve is prepared based on the exact sampling distribution. The plan parameters are determined by minimizing the average sample number while satisfying the quality levels demanded by both the producer and the consumer. The performance of the proposed plan is compared with the traditional variables single sampling plan (VSSP) and is examined in a case study.     

Design of an integrated quality assurance strategy in production systems

High-quality of products is a critical issue for manufacturers to maintain their competitiveness in global markets. For this reason, more attention has been paid by operations managers and academics to the design of quality assurance strategies, acceptance sampling plans and inspection allocation problems. In the last decades, international research has studied and introduced several models and approaches to investigate these issues. The purpose of this paper is to provide a new methodology for designing and selecting correct integrated quality assurance strategies, defining cost models for acceptance policies and inspection station configurations. Generally, high-quality of items is guaranteed by avoiding defects, mainly caused by non-conforming components, resulting from instantaneous and standard infant mortality. Thus, an optimal acceptance policy is defined in order to reduce the instantaneous infant mortality defects. A closed-form equation has been introduced to determine easily and quickly the optimal percentage of checked items. Furthermore, a more convenient inspection station configuration is determined in order to minimise the expected total cost, composed of testing, inspection and penalty cost elements. The innovative concept of defect rate as an inspection time variable dependent has been introduced. The impact of different economic and survival parameters on designing inspection policies is also investigated. Finally, a real-life case study demonstrates the applicability of this methodology in real production systems and several considerations are reported about the future research, that the authors will carry out.     

Minimum Variance and VOQL Sampling Plans

In this paper the outgoing quality and the total inspection under sampling for attributes are introduced as well-defined random variables. The variances of these random variables under total rectification are presented and are depicted graphically.

The thrust of the paper is to establish criteria for minimum variance sampling plans and lo present techniques for their determination. In addition to providing minimum variance sampling plans, a technique is developed for designing sampling plans with a designated maximum variance, a VOQL plan. The VOQL concept is analagous to the AOQL concept except in the VOQL plan, it is the variance which is maximized instead of the usual expected value of OQ.     

Designing two mixed quick switching sampling plans for linear profiles

This study compares two proposed mixed quick switching sampling (QSS) plans for linear profiles as the quality characteristic. For the QSS plans, we recommend a binomial attribute plan for normal inspection and then a variable sampling plan for tightened inspection based on capability index C_puA of linear profiles with one-sided specifications. The difference between the two proposed QSS plans is in the tightened inspection. Tightened inspection of the first proposed plan is a single sampling using C_puA index, but tightened inspection of the second plan is a multiple dependent state repetitive (MDSR) plan based on C_puA index. The optimal parameters are obtained by nonlinear optimization. Simulation study for selecting parameters is conducted with various combinations of specified acceptable quality level (AQL), limited quality level (LQL), producer's risk, and consumer's risk. Simulation results confirm that the second proposed QSS plan which applies variable MDSR at tightened inspection performs better than another proposed plan. Hence, the approach of the second proposed plan is demonstrated in an illustrative example.     

A resubmitted sampling scheme by variables inspection for controlling lot fraction nonconforming

The paper attempts to develop a resubmitted sampling scheme by variables inspection for controlling lot fraction nonconforming when the quality characteristic follows a normal distribution and has two-sided specification limits. In this paper, the plan parameters are determined by the classical two-point condition on the operating characteristic curve, which will satisfy the quality requirements and allowable risks by the producer and the consumer simultaneously. The behaviour of the proposed sampling plan is discussed and compared with the conventional single sampling plan by variables. The proposed plan requires smaller sample size for inspection with the same protection to the producer and the consumer especially when the quality of the submitted lot is good enough. Tables of the plan parameters under several selected quality requirements and risks are provided for practical applications, and the operating procedure is also presented and illustrated with an example.     

Optimum rectifying inspection plans

The rectifying inspection plan is a useful tool in statistical quality control (SQC) to assure the average product quality between several stages of a production line. Two optimum rectifying inspection plans are presented. They minimize either ATI (average total inspection) or AOQ (average outgoing quality) and use the other as a constraint. The definitions for both ATI and AOQ are generalized to handle the varying fraction defective of the incoming lots. A comprehensive study is conducted to compare the performance characteristics of the optimum rectifying inspection plans with several other plans. The results show that the optimum plans usually produce the best outcome and always satisfy the design specifications. Moreover, the results of a sensitivity study indicate that the optimum plans are fairly robust. Finally, a manufacturing example illustrates the applications and the managerial significance of the optimum rectifying inspection plans.     

Single Mixed Sampling Plan Based on Yield Index for Linear Profiles

Acceptance sampling plans have been widely used to decide whether an inspection lot from a supplier should be accepted or rejected. According to an economical point of view, a mixed sampling plan is better than the sampling plan by attributes. In some situations, lot sentencing can be determined by sampling plans by attributes and by variables simultaneously on the same product. In this paper, we propose a single mixed acceptance sampling plan based on the yield index for linear profiles for lot sentencing. The plan parameters are determined by minimizing sample size through a nonlinear optimization method such that the producer's risk and the consumer's risk are satisfied simultaneously for given values of acceptable quality level and limiting quality level. The results indicate that our proposed plan outperforms the single attributes sampling plan in terms of the sample size. One real example is used to illustrate the proposed method. Copyright © 2015 John Wiley & Sons, Ltd.     

A lots-dependent variables sampling plan considering supplier’s process loss and buyer’s stipulated specifications requirement

A well-established scheme and mechanism for deciding product acceptance is perceived as a win-win situation for the long-term supplier–buyer relationship. In this paper, we develop a lots-dependent variables sampling scheme for product acceptance determination. The dependent state is based on the sample information of the process capability index that incorporates the supplier’s process loss and the buyer’s demanded specifications requirement. This main scheme is implemented by a three-rule process that accepts or rejects a related lot conditional on the sample results of past lots. The plan-operational parameters satisfying the desired quality levels and constraining the supplier–buyer risks are determined by a non-linear optimisation model. In performance comparisons, our proposed plan demonstrated higher cost effectiveness and discriminatory power than the traditional variables single sampling plan. Finally, industrial applicability of our recommended sampling plans was investigated in a case study.     

The effects of inspection error on single sampling inspection plans

Acceptance sampling plans are designed under the assumption of perfect inspection. However, inspection tasks are not, even under ideal inspection conditions, free of error. In this paper we consider the effects of inspection error on probability of acceptance, average outgoing quality, and average total inspection. These measures are examined under both replacement and non -replacement assumptions. Also, a method is presented whereby an acceptance sampling plan may be designed which explicitly considers inspection error.     

Developing a variables repetitive group sampling scheme by considering process yield and quality loss

Acceptance sampling is a useful tool for determining whether submitted lots should be accepted or rejected. With the current increase in outsourcing production processes and the high-quality levels required, it is very desirable to have an efficient and economic sampling scheme. This paper develops a variables repetitive group sampling (RGS) plan that accounts for the process yield (meeting the manufacturing specifications) and the quality loss (variation from the target). The plan parameters are determined by solving a nonlinear optimisation problem. This implies that the plan parameters minimise the average sample number required for inspection and fulfil the classical two-point conditions on the operating characteristic (OC) curve. Besides, this paper investigates the efficiency of the proposed plan and compares it with the existing variables single sampling plan. Tables of the plan parameters for the proposed variables RGS plan are provided and an application example is presented for illustration.     

Variables Sampling Plan for Resubmitted Lots in a Process with Linear Profiles

Acceptance sampling plans include a sampling scheme and a set of rules for determining whether an inspection lot from a supplier should be accepted or rejected. In some circumstances, the supplier is allowed to resubmit lots for further inspection when the original inspection result is unacceptable. In this study, two variables sampling plans based on the process‐yield index for a process with linear profiles are proposed to deal with lot sentencing. The single sampling plan is a special case of the resubmitted lots sampling plan. The plan parameters are determined using a nonlinear optimization method under the given values of producer's risk, consumer's risk, acceptable quality level, and lot tolerance percent defective. Numerous tables are provided to determine the plan parameters. One real example is used to illustrate our proposed method. Copyright © 2015 John Wiley & Sons, Ltd.     

Multiattribute Bayesian Acceptance Sampling Plans for Screening and Scrapping Rejected Lots

A general model for multiattribute Bayesian acceptance sampling plans is developed which incorporates the multiattribute utility function of a decision maker in its design. The model accommodates various dispositions of rejected lots such as screening and scrapping. The disposition of rejected lots is shown to have a substantial impact on the solution approach used and on the ease of incorporation of multiattribute utility functions in terms of their measurement complexity, functional form, and parameter estimation. For example, if all attributes are screenable upon rejection, and the prior distributions of lot quality on each attribute are independent, then an optimal multiattribute sampling plan can be obtained simply by solving for an optimal single sampling plan on each attribute independently. A discrete search algorithm, based on pattern search, is also developed and shown to be very effective in obtaining an optimal multiattribute inspection plan when such separability cannot be accomplished.     

A Single Sampling Plan Based on Exponentially Weighted Moving Average Model for Linear Profiles

The exponentially weighted moving average (EWMA) model has been successfully used in acceptance sampling plans. The EWMA model provides the quality information of the current lot and the preceding lots. In addition, a multiple dependent state (MDS) sampling plan considers the quality information of the preceding lots. In this study, we present two new sampling plans for linear profiles. One is based on EWMA model with yield index using the single sampling plan, and the other is based on EWMA model with yield index using the MDS sampling plans. The plan parameters are determined by a nonlinear optimization approach. As the smoothing parameter value equals to one, the first proposed plan becomes the traditional single sampling plan. In addition, we compare the proposed plans with the traditional single sampling plan. The results indicate that the MDS sampling plan based on EWMA model with yield index with smaller value of smoothing parameter performs better than the traditional single sampling plan and the single sampling plan based on EWMA model with yield index in terms of the sample size required. One real example is used to illustrate the proposed plan. Copyright © 2015 John Wiley & Sons, Ltd.     

Tools and Methods for the Improvement of Quality

In developing quality-control procedures, a step-loss function has been used implicitly or explicitly to describe consumer perceptions about product quality. A quadratic loss function has been suggested by Taguchi as an alternative to the step-loss function in measuring the loss due to imperfect product quality (cost of acceptance). In this article, Bayesian analyses of the known-standard-deviation acceptance-sampling problem are described for both the step and quadratic loss functions with three cost components—cost of inspection, cost of acceptance, and cost of rejection. A normal prior distribution is used for the lot mean. Efficient procedures for finding minimum expected cost procedures are given. For a particular example, comparisons are made of how optimal sampling plans and costs computed under the two cost structures change as the form of the prior distribution and misspecification of its mean and variance are varied. Sensitivity analyses for both cost functions show that the optimal sampling plan is robust with respect to the form of the prior distribution, as well as to misspecification of its mean and variance, if the tail specification reasonably approximates that of a normal distribution.