Selection of csp-t plans

Various continuous sampling plans have been proposed for monitoring the quality of continuous production processes. The multi-level continuous sampling plan of MIL-STD-1235C (1988) is designated as CSP-T Plan. CSP-T plan is a three-level tightened continuous sampling plan. It requires a switch to 100% inspection, at any level, whenever a nonconforming unit is found. This provides quick rectification in the event of a shift in quality. In this paper certain performance characteristics of CSP-T plan are derived using the approach of Stephens (1979) under the assumption that the production process is in statistical control. For the selection of CSP-T plans, two tables are given. These tables can be used to obtain parameters i (clearance interval) and f (sampling fraction) of the CSP-T plan for given acceptable quality level (AQL) with producer’ risk α=0.05 or the limiting quality level(LQL) with consumer’ risk β=0.10 and the outgoing quality limit(AOQL). Two examples are also given to illustrate the selection of plans from these tables     

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.     

Tightened single-level continuous sampling plan

In this paper, a new tightening concept has been incorporated into the single-level continuous sampling plan CSP-1, such that quality degradation will warrant sampling inspection to cease beyond a certain number of sampled items, until new evidence of good quality is established. The expressions of the performance measures for this new plan, such as the operating characteristic, average outgoing quality and average fraction inspected, are derived using a Markov chain model. The advantage of the tightened CSP-1 plan is that it is possible to lower the average outgoing quality limit.     

Developing a variables two-plan sampling system for product acceptance determination

In this paper, a variables tightened-normal-tightened (TNT) two-plan sampling system based on the widely used capability index C_pk is developed for product acceptance determination when the quality characteristic of products has two-sided specification limits and follows a normal distribution. The operating procedure and operating characteristic (OC) function of the variables TNT two-plan sampling system, and the conditions for solving plan parameters are provided. The behavior of OC curves for the variables TNT sampling system under various parameters is also studied, and compared with the variables single tightened inspection plan and single normal inspection plan.     

Selection of tightened two-level continuous sampling plans

Lieberman and Solomon (1955) introduced multi-level continuous sampling plans and Derman et al. (1957) extended them as tightened multi-level continuous plans. MLP-T plan is one of the three tightened multi-level continuous sampling plans of Derman et al. (1957). In this paper, we restrict our discussion to MLP-T plans with two sampling levels. Using a Markov chain model, expressions for the average outgoing quality, the average fraction inspected and the operating characteristic function are derived. Four tables are given to enable selection of MLP-T plans with two sampling levels when the acceptable quality level or the limiting quality level and the average outgoing quality limit are specified.     

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     

Economic design of continuous sampling plan under linear inspection cost

The article explores the problem of an economically based type I continuous sampling plan (CSP-1 plan) under linear inspection cost. By assuming that the per unit inspection cost is linearly proportional to the average number of inspections per inspection cycle, and by solving the modified Cassady et al.'s model, we not only have the required level of product quality but also obtain the minimum total expected cost per unit produced.     

Designing and selection of two-plan variables scheme indexed by crossover point

In this paper, the scheme of the inspection plan, namely the tightened normal tightened (n_T, n_N; k) is considered and procedures and necessary tables are developed for the selection of the variables sampling scheme, indexed through crossover point (COP). The importance of COP, the properties and advantages of the operating characteristic curve with respect to COP are studied.     

Minimum average fraction inspected for TCSP-1 plan

This paper presents the calculation of the average outgoing quality limit (AOQL) for the tightened single-level continuous sampling plan (TCSP-1 plan) based on a numerical method. A solution procedure is developed to find the parameters (i, f, k) that will meet the AOQL requirement, while also minimizing the average fraction inspected (AFI) for the TCSP-1 plan when the process average [pbar] (> AOQL) is known.     

Design and evaluation of skip-lot sampling plans of type SkSP-3

This paper presents a system of skip-lot sampling inspection plans designated as SkSP-3 based on the principle of a continuous sampling plan of type CSP-2. Expressions for performance measures such as Operating Characteristic function and ASN function are derived by the Markov chain approach. Selection of SkSP-3 with a single sampling plan having acceptance number zero as the reference plan is discussed.     

On a problem in two-stage sampling

Beattie 1962 proposed an alternate procedure for monitoring continuous processes which, unlike existing continuous samplingplans under Mil-Std-1235B, does not require periods of 100% inspection. The procedure, which is a composite of two Cumulative (Cusum) procedures, will be analyzed in terms of Cusum average run length (ARL) performance. An algorithm is proposed for generating sample plans based upon specifications on ARL performance in each of the Cusum sampling zones and the overall sampling risks of the procedure. The algorithm is fully described, and implemented using exact expressions for the Cusum ARL. An illustrated example is provided, describing how a continuous sampling maybe selected which matches performances about the AQL for a related plan in ISO 3951.     

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.     

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.     

Combined continuous lot by lot acceptance sampling plan

For production processes involving low fraction non-conforming, the sample sizes of the usual attribute inspection plans are very large. A continuous sampling plan for such processes would also require either a large clearance interval or a large sampling fraction. This paper simplifies the approach of combining the lot by lot and continuous sampling plans recommended by Pesotchinsky (1987) and provides various performance measures for the combined plan. A discussion of the choice of the parameters is also given.     

Design and Construction of a Variables Repetitive Group Sampling Plan for Unilateral Specification Limit

This paper attempts to develop a repetitive group sampling (RGS) plan by variables inspection for controlling the process fraction defective or the number of nonconformities when the quality characteristic follows a normal distribution and has only the lower or upper specification limit. The proposed sampling plan is derived by the exact sampling distribution rather than the approximation approach. The plan parameters are solved by a nonlinear optimization model which minimizes the average sample number required for inspection and fulfills the classical two-point conditions on the operating characteristic (OC) curve. The efficiency of the proposed variables RGS is examined and also compared with the existing variables single sampling plan in terms of the sample size required for inspection. The results indicate that the proposed variables RGS plan could significantly reduce samples required for inspection compared to the traditional variables single sampling plan.     

Average outgoing quality of CSP-C continuous sampling plan under short run production processes

A CSP-C continuous sampling plan is a new single-level continuous sampling procedure developed by Govindaraju & Kandasamy (2000) by incorporating the concept of acceptance number to the CSP-1 plan for the application of continuous production processes. In this new plan, the sampling inspection phase is characterized by a maximum allowable number of non-conforming units, c, and a constant sampling rate, f. Govindaraju & Kandasamy (2000) derived the performance measures such as average outgoing quality (AOQ), average fraction inspected (AFI) etc, of the CSP-C plan using a Markov chain model for long run production processes. Yang (1983) has observed that the AOQ and AFI, being long run average measures, are not satisfactory measures of performance for short run production processes. Hence, formulas are derived in this paper, using the renewal theory approach enabling one to compute AOQ and AFI for both long run and short run production processes. Numerical illustrations are also given. By simulation, the accuracy of the short run measures is studied.     

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.     

Repetitive group sampling procedure for variables inspection

This paper introduces the concept of repetitive group sampling (RGS) for variables inspection. The repetitive group sampling plan for variables inspection will be useful when testing is costly and destructive. The advantages of the variables RGS plan over variables single sampling plan, variables double sampling plan and attributes RGS plan are discussed. Tables are also constructed for the selection of parameters of known and unknown standard deviation variables repetitive group sampling plan indexed by acceptable quality level and limiting quality level.     

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.     

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.