Developing of a new system of skip-lot reinspection schemes

The skip-lot sampling plans are widely used in industries for quality inspection of products in order to reduce the sampling costs and inspection efforts when products have good quality history. Also, the skip-lot sampling plan concept is sound and useful and it is economically advantageous to use the skip-lot approach in the design of sampling plans. Thus, the skip-lot sampling plans are useful to minimize the cost of the inspection particularly in costly and destructive testing. Hence, a new system of skip-lot sampling plans designated as SkSP-2-R plan is developed in this article by incorporating the idea of resampling procedure in the skip-lot sampling plans of type SkSP-2. A Markov chain formulation and derivation of performance measures for this new plan are presented. The properties and advantages of the SkSP-2-R plan are studied with single sampling plan as the reference plan. The response-to-change characteristics of the SkSP-2-R plan are also investigated, based on the average run length.     

Optimal designing of two-level skip-lot sampling reinspection plan

Skip-lot sampling plan is often applied in industries for reducing the cost and effort of the inspection of the product having excellent quality history. Consequence of skip-lot sampling plans is to reduce the cost of inspection so which are more attractive in economical aspect. In this paper, we develop a sampling plan by incorporating the idea of resampling in two-level skip lot sampling plan and the new plan is designated as SkSP-2L.1-R. This paper presents the Markov chain formulation of the proposed plan along with the derivation of performance measures of the plan. We also provide the designing methodology to determine the optimal parameters of the SkSP-2L.1-R plan so as to minimize the average sample number by using two points on the operating characteristic curve approach. By contemplating various combinations of producer and consumer quality levels along with respective risks, a table is constructed to determine the optimal parameters. An industrial application of the proposed SkSP-2L.1-R plan is discussed. The SkSP-2L.1-R with single sampling plan as a reference plan is compared with the conventional single sampling plan, SkSP-2 plan and SkSP-2-R plan and proved that the proposed SkSP-2L.1-R plan outperforms these plans.KEYWORDS: Consumer quality level, average sample number, producer quality level, resampling scheme, skip-lot sampling     

Optimal designing of skip-lot sampling plan of type SkSP-2 with group acceptance sampling plan as reference plan under Burr-type XII distribution

In this paper, skip-lot sampling plan of type SkSP-2 with group acceptance sampling plans is proposed when the lifetime of the product follows the Burr-type XII distributions. The optimal parameters of the proposed plan are determined when two points on operating characteristics curve namely acceptable quality level and limiting quality level and the number of testers are specified. We also considered the Burr-type XII distribution to find the plan parameters. Several tables are given for practical use. We compare the results of proposed plan with the existing plans. Results are explained using real-world examples.     

Efficient Bayesian sampling plans for exponential distributions with random censoring

This paper considers Bayesian sampling plans for exponential distribution with random censoring. The efficient Bayesian sampling plan for a general loss function is derived. This sampling plan possesses the property that it may make decisions prior to the end of the life test experiment, and its decision function is the same as the Bayes decision function which makes decisions based on data collected at the end of the life test experiment. Compared with the optimal Bayesian sampling plan of Chen et al. (2004), the efficient Bayesian sampling plan has the smaller Bayes risk due to the less duration time of life test experiment. Computations of the efficient Bayes risks for the conjugate prior are given. Numerical comparisons between the proposed efficient Bayesian sampling plan and the optimal Bayesian sampling plan of Chen et al. (2004) under two special decision losses, including the quadratic decision loss, are provided. Numerical results also demonstrate that the performance of the proposed efficient sampling plan is superior to that of the optimal sampling plan by Chen et al. (2004).     

Optimal designing of a new mixed variable lot-size chain sampling plan based on the process capability index

In this paper, a new mixed sampling plan based on the process capability index (PCI) C_pk is proposed and the resultant plan is called mixed variable lot-size chain sampling plan (ChSP). The proposed mixed plan comprises of both attribute and variables inspections. The variable lot-size sampling plan can be used for inspection of attribute quality characteristics and for the inspection of measurable quality characteristics, the variables ChSP based on PCI will be used. We have considered both symmetric and asymmetric fraction non conforming cases for the variables ChSP. Tables are developed for determining the optimal parameters of the proposed mixed plan based on two points on the operating characteristic (OC) approach. In order to construct the tables, the problem is formulated as a non linear programming where the average sample number function is considered as an objective function to be minimized and the lot acceptance probabilities at acceptable quality level and limiting quality level under the OC curve are considered as constraints. The practical implementation of the proposed mixed sampling plan is explained with an illustrative real time example. Advantages of the proposed sampling plan are also discussed in terms of comparison with other existing sampling plans.     

Designing of a mixed-chain sampling plan based on the process capability index with chain sampling as the attributes plan

In this article, we propose a new mixed chain sampling plan based on the process capability index C_pk, where the quality characteristic of interest having double specification limits and follows the normal distribution with unknown mean and variance. In the proposed mixed plan, the chain sampling inspection plan is used for the inspection of attribute quality characteristics. The advantages of this proposed mixed sampling plan are also discussed. Tables are constructed to determine the optimal parameters for practical applications by formulating the problem as a non linear programming in which the objective function to be minimized is the average sample number and the constraints are related to lot acceptance probabilities at acceptable quality level and limiting quality level under the operating characteristic curve. The practical application of the proposed mixed sampling plan is explained with an illustrative example. Comparison of the proposed sampling plan is also made with other existing sampling plans.     

An improved sampling plan by variables inspection with consideration of process yield and quality loss

Repetitive group sampling (RGS) plan, a modified version of single sampling (SS) plan, has been shown to be more efficient than the SS plan for lot sentencing. However, because it does not consider the valuable sample information from preceding lots, that could reduce its sampling efficiency and discriminatory power. Therefore, this study proposes a modified-RGS plan by considering the quality history of preceding lots based on the C_pmk index. Additionally, the mathematical model for the plan parameters is formulated such that the objective function is to minimize the average sample number (ASN), and the required quality levels and risks as specified by producer and consumer are satisfied. The performance of the proposed plan is examined and compared with traditional sampling plans. Finally, an example is presented to illustrate its applicability.     

Double and group acceptance sampling plan for truncated life test based on inverse log-logistic distribution

This paper introduces a double and group acceptance sampling plans based on time truncated lifetimes when the lifetime of an item follows the inverse log-logistic (ILL) distribution with known shape parameter. The operating characteristic function and average sample number (ASN) values of the double acceptance sampling inspection plan are provided. The values of the minimum number of groups and operating characteristic function for various quality levels are obtained for a group acceptance sampling inspection plan. A comparative study between single acceptance sampling inspection plan and double acceptance sampling inspection plan is carried out in terms of sample size. One simulated example and four real-life examples are discussed to show the applicability of the proposed double and group acceptance sampling inspection plans for ILL distributed quality parameters.     

Multiple dependent state repetitive sampling plans with or without auxiliary variable

In this article, a new variable acceptance sampling plan has been developed using the multiple dependent state repetitive sampling scheme for the normal distribution. The plan parameters have been determined so as to minimize the average sample number while satisfying the producer's and the consumer's risks under the operating characteristic function. The multiple dependent state repetitive sampling scheme has also been extended to the case of utilizing an auxiliary variable. The proposed sampling plans are compared with the existing sampling plans.     

A new mixed chain sampling plan based on the process capability index for product acceptance

This article proposes a new mixed chain sampling plan based on the process capability index C_pk, where the quality characteristic of interest follows the normal distribution with unknown mean and variance. The advantages of this proposed mixed sampling plan are also discussed. Tables are constructed to determine the optimal parameters for practical applications. In order to construct the tables, the problem is formulated as a nonlinear programming where the objective function to be minimized is the average sample number and the constraints are related to lot acceptance probabilities at acceptable quality level and limiting quality level under the operating characteristic curve. The practical application of the proposed mixed sampling plan is explained with an illustrative example. Comparison of the proposed sampling plan is also made with other existing sampling plans.     

Curtailed Bayesian sampling plans for exponential distributions based on Type-II censored samples

This paper studies the problem of designing a curtailed Bayesian sampling plan (CBSP) with Type-II censored data. We first derive the Bayesian sampling plan (BSP) for exponential distributions based on Type-II censored samples in a general loss function. For the conjugate prior with quadratic loss function, an explicit expression for the Bayes decision function is derived. Using the property of monotonicity of the Bayes decision function, a new Bayesian sampling plan modified by the curtailment procedure, called a CBSP, is proposed. It is shown that the risk of CBSP is less than or equal to that of BSP. Comparisons among some existing BSPs and the proposed CBSP are given. Monte Carlo simulations are conducted, and numerical results indicate that the CBSP outperforms those early existing sampling plans if the time loss is considered in the loss function.     

Two-Stage Variables Acceptance Sampling Plans Using Process Loss Functions

In this article, a variable two-stage acceptance sampling plan is developed when the quality characteristic is evaluated through a process loss function. The plan parameters of the proposed plan are determined by using the two-point approach and tabulated according to various quality levels. Two cases are discussed when the process mean lies at the target value and when it does not, respectively. Extensive tables are provided for both cases and the results are explained with examples. The advantage of the proposed plan is compared with the existing variable single acceptance sampling plan using the process loss function.     

A flexible sampling scheme for variables inspection with loss consideration

This article develops a variables sampling scheme for resubmitted lots by incorporating the concept of Taguchi loss function. The probability of lot acceptance is derived based on the exact sampling distribution and two-point condition on operating characteristic curve is used to determine the plan parameters that meet both the producer's and consumer's quality and risk requirements. Moreover, the performance of the proposed variables resubmitted sampling plan is investigated and compared with the classical variables single sampling plan. The results indicate that the developed resubmitted sampling plan can provide the same protection with less inspection when the submitted lot is good enough. Tables of the plan parameters under various conditions are provided and the use of the proposed plan is also illustrated with an example.     

Exact Bayesian Variable Sampling Plans for the Exponential Distribution Based on Type-I and Type-II Hybrid Censored Samples

A sampling plan with a polynomial loss function for the exponential distribution is considered. From the distribution of the maximum likelihood estimator of the mean of an exponential distribution based on Type-I and Type-II hybrid censored samples, we obtain an explicit expression for the Bayes risk of a sampling plan with a quadratic loss function. Some numerical examples and comparisons are given to illustrate the effectiveness of the proposed method, and a robustness study reveals that the proposed optimal sampling plans are quite robust.     

Attribute-variable Inspection Policy for Lots Using Resampling Based on EWMA

A new mixed sampling plan which is a combination of the attribute single sampling plan and variables resampling scheme based on EWMA statistic is proposed in this paper. The operating characteristic function of the proposed plan is derived and the plan parameters are determined such that the probability of acceptance of good lot is larger than the specified producer's confidence level and the bad lot acceptance probability is smaller than the consumer's confidence level. The efficiency and the advantages of the proposed plan are discussed over the existing attribute sampling plan. The extensive tables are provided for industrial applications. The use of tables is discussed with the help of a real-time industrial example.     

Stratified Double Sampling with Continuous Outcomes: Design and Analysis

We provide a method for estimating the sample mean of a continuous outcome in a stratified population using a double sampling scheme. The stratified sample mean is a weighted average of stratum specific means. It is assumed that the fallible and true outcome data are related by a simple linear regression model in each stratum. The optimal stratified double sampling plan, i.e. , the double sampling plan that minimizes the cost of sampling for fixed variances, or alternatively, minimizes the variance for fixed costs, is found and compared to a standard sampling plan. The design parameters are the total sample size and the number of doubly sampled units in each stratum. We show that the optimal double sampling plan is a function of the between-strata and within-strata cost and variance ratios. The efficiency gains, relative to standard sampling plans, under broad set of conditions, are considerable.     

Optimal Designing of Variables Sampling Plan for Resubmitted Lots

This article proposes a variables sampling plan that can be applied for sampling inspection of resubmitted lots when the quality characteristic of interest follows the normal distribution. Resubmission of lots for inspection is allowed in some situations where the original inspection results are suspected or when the supplier or producer is allowed to opt for resampling as per the provisions of the contract, etc. The advantages of this proposed variables sampling plan over the existing single sampling variables plan are discussed. Tables are also constructed for the selection of optimal parameters of known and unknown standard deviation variables resampling scheme for specified two points on the operating characteristic curve, namely, the acceptable quality level and the limiting quality level along with the producer and consumer's risks. The optimization problem is formulated as a nonlinear programming where the objective function to be minimized is the average sample number and the constraints are related to lot acceptance probabilities at acceptable quality level and limiting quality level under the operating characteristic curve.     

Optimal designing of an SkSP-R double sampling plan

In this article, we propose a method of planning and determining the optimum parameters of a SkSP-R skip-lot sampling plan by using the attribute double sampling plan as the reference plan. The SkSP-R plan is a new type of skip-lot sampling plan which has a provision for re-inspecting the submitted lots. The optimal plan parameters of the suggested sampling plan are estimated with the target that the average sample number be minimized and satisfying both the specified producer's as well as the consumer's risks simultaneously. In order to obtain the optimum parameters, tables are also built for different combinations of the acceptable quality level and the limiting quality level in conjunctions with different producer's and consumer's risks. An illustrative example is provided for the implementation of the suggested plan. The advantages of the suggested plan over the existing conventional sampling plans and other existing skip-lot sampling plans are also described.     

Optimal Bayesian variable sampling plans for exponential distributions based on modified type-II hybrid censored samples

For the conventional type-II hybrid censoring scheme (HCS) in Childs et al., a Bayesian variable sampling plan among the class of the maximum likelihood estimators was derived by Lin et al. under the loss function, which does not include the cost of experimental time. Instead of taking the conventional type-II hybrid censoring scheme, a persuasive argument leads to taking the modified type-II hybrid censoring scheme (MHCS) if the cost of experimental time is included in the loss function. In this article, we apply the decision-theoretic approach for the concerned acceptance sampling. With the type-II MHCS, based on a sufficient statistics, the optimal Bayesian sampling plan is derived under a general loss function. Furthermore, for the conjugate prior distribution, the closed-form formula of the Bayes decision rule can be obtained under the quadratic decision loss. Numerical study is given to demonstrate the performance of the proposed Bayesian sampling plan.     

Modified CSP-C Continuous Sampling Plan for Consumer Protection

Dodge (1943) introduced a single level attribute continuous sampling plan designated as CSP-1 for the application of continuous production processes. Govindaraju & Kandasamy (2000) developed a new single level continuous sampling plan whose sampling inspection phase is characterized by a maximum allowable number of non-conforming units c, and a constant sampling rate f and was designated as CSP-C. In this paper, a modification is proposed on the CSP-C continuous sampling plan. In this modified plan, sampling inspection is continued until the occurrence of c+1 non-conforming units, provided the first m sampled units have been found conforming during the sampling phase. Using a Markov chain model, expressions for the performance measures of the modified CSP-C plan are derived. The main advantage of the modified plan is that it is possible to lower the average outgoing quality limit.