Two-point Condition Research Articles

A cost-effective skip-lot sampling scheme using loss-based capability index for product acceptance determination

Acceptance sampling plans are essential for quality control and assurance, empowering businesses to make well-informed decisions regarding lot disposition. Among these plans, the skip-lot sampling plan (SkSP) stands out as a cost-effective strategy for suppliers with consistently excellent product quality. In this article, we present an enhanced version of the SkSP, termed Cpm-based SkSP-RGSP, which incorporates a repetitive group sampling plan (RGSP) using the Taguchi capability index Cpm as its reference plan. We develop an optimization model to ensure that the proposed plan meets a two-point condition in its operating characteristic function and effectively minimizes the average sample number (ASN). Through investigations of two critical performance measures, ASN and discriminatory power, we demonstrate the superiority of our plan over conventional sampling approaches. Detailed plan parameters, including required sample size and lot sentencing criteria, are provided to assist examiners in practical implementation. Additionally, we showcase the plan's applicability and practicality through a case study. This research contributes to the field of quality control by offering an efficient and cost-effective sampling scheme for lot disposition.

MODIFICATION OF THE PARAMETRIZATION METHOD FOR SOLVING A BOUNDARY VALUE PROBLEM FOR LOADED DIFFERENTIAL EPCAG

In this paper, modification of Dzhumabaev parameterization method is developed to a boundary value problem for systems of loaded differential equations with piecewise constant argument of generalized type (EPCAG). The method is based on reducing the considering problem to an equivalent multi-point boundary value problem for ordinary differential equations with parameters. An equivalent boundary value problem with parameters consists of the Cauchy problem for a system of ordinary differential equations with parameters, a two-point condition, a continuity condition, and additional conditions for a piecewise constant argument. The solution of the Cauchy problem for a system of ordinary differential equations with parameters is constructed using the fundamental matrix of the differential equation. A system of linear algebraic equations for the parameters is compiled using the values of the solution at the corresponding points and substituting them into the two-point condition, the continuity condition, and the conditions for the piecewise constant argument. A modification of Dzhumabaev parameterization method for solving the considering boundary value problem is proposed which is based on solving the constructed system and the 4th order Runge-Kutta method for solving the Cauchy problem on subintervals. The obtained results are verified by a numerical example. Numerical analysis showed high efficiency of the constructed modification of Dzhumabaev parameterization method.

Construction and Selection of Double Inspection Single Sampling Plan for an Independent Process Using Bivariate Poisson Distribution

There are more sampling concepts active in production Industries, for inspecting the samples and analysing performance of the population. Also the sampling plans reduce errors in the production and produce the error free products. In this study, construction and selection of Double Inspection with reference to Single Sampling Plan i.e., DISSP, by attribute are investigated by using the Bivariate Poisson distribution. The Methodology, DISSP, was proposed based on two quality characteristics of the same sample size, and the planning parameters (n, C₁, C₂) are based on the operating characteristics, the conventional two-point condition by the planning table parameters (AQL and LQL). It is based on selected quality requirements and risks designed to allow manufacturers to easily determine the required sample size and corresponding acceptance criteria. A Comparison was done based on the efficiency of the plan with an existing single sampling plan and gave a numerical example to expose the operating tables. Also, the study shows the advantages of the proposed plan, and performance of the curves like, Operating characteristics, Average Outgoing Quality, and Average Total Inspection to expose the proposed double inspection sampling plan.

On a nonlinear differential equation with two-point nonlocal condition with parameters

On a nonlinear differential equation with two-point nonlocal condition with parameters

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.

A Sampling Scheme for Resubmitted Lots Based on One-Sided Capability Indices

Acceptance sampling plans provide decision rules for lot acceptance determination based on the required quality levels and allowable risks. In this paper, we develop a variables sampling plan based on the one-sided process capability indices, CPU and CPL, when lot resubmissions are permitted on non-acceptance of the original inspection. The plan parameters are determined by solving two non-linear equations simultaneously and fulfill the two-point condition on the operating characteristic (OC) curve. The OC curve of the proposed plan is derived based on the exact sampling distribution rather than approximation. Moreover, the behavior of the proposed resubmitted sampling plan for various parameters is examined and discussed. For practical purposes, the sample size and critical acceptance value are tabulated for various parameter values. An example is used to demonstrate the implementation of the proposed resubmitted sampling plan.

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.

Developing a sampling plan by variables inspection for controlling lot fraction of defectives

Acceptance sampling has been widely used tool for determining whether the submitted lot should be accepted or rejected. However, it cannot avoid two kinds of risks, accepting undesired poor product lots and rejecting good product lots. Such risks are even more significant as the rapid advancement of the manufacturing technology and stringent customers demand is enforced. A yield index Spk has been developed to provide an exact measure on process yield or fraction nonconforming for normally distributed processes with two-sided specification limits. Therefore, the aim of this paper is to develop a variables sampling plan for evaluating the lot or process fraction nonconforming based on the yield index. The probability of lot acceptance is derived based on the sampling distribution and two-point condition on OC curve is used to determine the plan parameters. Tables of the plan parameters and step-by-step procedure are provided for the practitioner to make decision on lot sentencing especially for situations of products with very low fraction of nonconformities.

On exact and optimal single-sampling plans by variables

We deal with sampling by variables with two-way protection in the case of a \(N\>(\mu ,\sigma ^2)\) distributed characteristic with unknown \(\sigma \). The LR sampling plan proposed by Lieberman and Resnikoff (JASA 50: 457\({-}\)516, 1955) and the BSK sampling plan proposed by Bruhn-Suhr and Krumbholz (Stat. Papers 31: 195–207, 1990) are based on the UMVU and the plug-in estimator, respectively. For given \(p_1\) (AQL), \(p_2\) (RQL) and \(\alpha ,\beta \) (type I and II errors) we present an algorithm allowing to determine the optimal LR and BSK plans having minimal sample size among all plans satisfying the corresponding two-point condition on the OC. An R (R: A language and environment for statistical computing, R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0, URL http://www.R-project.org/2012) package, ExLiebeRes‘ (Krumbholz and Steuer ExLiebeRes: calculating exact LR- and BSK-plans, R-package version 0.9.9. http://exlieberes.r-forge.r-project.org2012) implementing that algorithm is provided to the public.

Problem with nonlocal two-point condition in time for a homogeneous partial differential equation of infinite order with respect to space variables

We specify a class of unique solvability of a problem with nonlocal boundary condition for a homogeneous partial differential equation of the first order with respect to time and of infinite order with respect to space variables with constant complex coefficients. In the class of quasipolynomials of special form, we indicate formulas for the construction of a solution of the problem that require a finite number of differentiation operations of analytically given functions. For the case where there exists a nonunique solution of the problem, we present an algorithm of the construction of its partial solution.

A two-point condition for characterizing the exponential distribution by means of identifically distributed properties related to order statistics

Let X1, X2, ..., Xn be independent and identically distributed random variables subject to a continuous distribution function F. Let X1∶n, X2∶n, ..., Xn∶n denote the corresponding order statistics. Write $$P(X_{k + s:n} - X_{k:n} \geqslant x) = P(X_{s:n - k} \geqslant x),$$ (*) where n, k are fixed integers. We apply a result of Marsaglia and Tubilla on the lack of memory of the exponential distribution finction assuming that certain distribution functions involving the above order statistics are equal in two incommensurable points τ1, τ2 > 0; this characterizes the exponential distribution. As a special case it turns out that the equality (*) assumed for s=1, 2 and x=τ1, τ2 implies that F is exponential.

Well-posed problems in a layer with differential operators in boundary conditions

We obtain a criterion for the well-posedness of a boundary-value problem in the layer ℝnx [0, T] for a linear differential evolution equation with constant complex coefficients in the class of functions of power growth under a two-point condition that contains two arbitrary differential operators with respect to space variables (one of the operators can be zero). Examples of well-posed and ill-posed problems of this form are given.

Some new fixed point theorems of increasing operators and applications

Let E be a Banach space, P a cone in E, A:P→P an increasing operator. If there exist xO, yOeP such that xO⩽AxO, AyO⩽yO and xO⩽yO, then we say A satisfies two-point extension condition. We prove some fixed point theorems under two'point extension condition, and give some applications to nonlinear differential equations.

An initial value problem for a functional equation

We consider the functional equationf(A(x,y))=B(f(x),f(y)), whereA andB are averages. It is known that such a functional equation has exactly one continuous solution satisfying a given two-point condition. By analogy with the theory of differential equations, we may regard the functional equation, together with a two-point condition, as a boundary value problem. (Then each boundary value problem has a unique continuous solution.) If we replace the two-point condition with the specification of a value and derivative at just one point, we obtain an initial value problem.

Topological Group Foundations of Rigid Space Geometry

1. Hilbert, after building up geometry from a point of view which relegated considerations to the background [4], built up plane geometry afresh [5] on the foundation of groups of homeomorphisms of the number plane, both continuity concepts. It is this latter point of view with which we are concerned in this paper. Hilbert carried out this program only for the plane but he hinted that it might be possible to carry it out in a somewhat similar way for three-space. Kerekjarto took up the problem [6] for threespace and made a great deal of progress with it, but, as he wrote before the recent developments in topological groups, he found it necessary to employ a stronger set of axioms than is necessary now. Relying on the theory of topological groups we recently characterized the rotation group of three-space [9], and in commenting on that work P. A. Smith [12] suggested that it might provide the basis for an extension of Hilbert's program to three-space. The purpose of this paper is twofold. In the first place we shall characterize the classical space geometries on the basis of a fairly weak set of axioms, and in the second place we shall show that Hilbert's axioms for the case of the plane can be weakened by replacing what might be called his three-point condition by a two-point condition. We achieve this latter purpose more or less incidentally to the first. In comparing our set of three-space axioms with Hilbert's axfoms for the plane we find that the first axiom is the same in both cases. The third axiom of this paper is weaker than Hilbert's, and our second purpose above is to show that this weaker axiom also suffices in Hilbert's case. Our second axiom is weaker than Hilbert's second axiom in that it relates to the subgroup leaving a single point fixed, but it is incomparably stronger in what it asks of that one subgroup. We do not, in this paper, settle the question of whether or not Hilbert's second axiom is adequate for three-space geometries. This question is bound up with an unsolved problem concerning transformation groups. Finally we wish to remark that instead of assuming that the space we are dealing with is ordinary three-space, it is only necessary to make certain topological assumptions on the space from which it follows by virtue of the

Topological group foundations of rigid space geometry

1. Hilbert, after building up geometry from a point of view which relegated continuity considerations to the background [4], built up plane geometry afresh [5] on the foundation of groups of homeomorphisms of the number plane, both "continuity" concepts. It is this latter point of view with which we are concerned in this paper. Hilbert carried out this program only for the plane but he hinted that it might be possible to carry it out in a somewhat similar way for three-space. Kerekjarto took up the problem [6] for threespace and made a great deal of progress with it, but, as he wrote before the recent developments in topological groups, he found it necessary to employ a stronger set of axioms than is necessary now. Relying on the theory of topological groups we recently characterized the rotation group of three-space [9], and in commenting on that work P. A. Smith [12] suggested that it might provide the basis for an extension of Hilbert's program to three-space. The purpose of this paper is twofold. In the first place we shall characterize the classical space geometries on the basis of a fairly weak set of axioms, and in the second place we shall show that Hilbert's axioms for the case of the plane can be weakened by replacing what might be called his "three-point condition" by a two-point condition. We achieve this latter purpose more or less incidentally to the first. In comparing our set of three-space axioms with Hilbert's axfoms for the plane we find that the first axiom is the same in both cases. The third axiom of this paper is weaker than Hilbert's, and our second purpose above is to show that this weaker axiom also suffices in Hilbert's case. Our second axiom is weaker than Hilbert's second axiom in that it relates to the subgroup leaving a single point fixed, but it is incomparably stronger in what it asks of that one subgroup. We do not, in this paper, settle the question of whether or not Hilbert's second axiom is adequate for three-space geometries. This question is bound up with an unsolved problem concerning transformation groups. Finally we wish to remark that instead of assuming that the space we are dealing with is ordinary three-space, it is only necessary to make certain topological assumptions on the space from which it follows by virtue of the