Process Capability Evaluation for the Process of Product Families

Process capability analysis is a highly effective means of assessing the process ability of manufacturing product that meets specifications. A larger process capability index (PCI) implies a higher process yield, and lower expected process loss. Many PCIs have been effectively and widely used to determine whether the quality of a process meets preset targets. However, those existing PCI cannot be applied to evaluate the process capability of a process producing a product family. This work presents a novel PCI (C _pm ^T ) which takes into account all family members and obtains the probability density function for the PCI for a product family. The relationship between the PCI and process yield is described. An example is provided to demonstrate the methodology for practical application.     

Control Charts for One-sided Capability Indices

There are many industrial product characteristics are desired to be the bigger the best and the smaller the best. The two well-know processes capability indices C _pl and C _pu, which measure larger-the-better and smaller-the-better process capabilities. Obviously, the formulae for the two indices C _pl and C _pu are easy to understand and straightforward to apply. Thus, indices C _pl and C _pu have been utilized by a number of Japanese companies and the U.S. automotive industry by Ford Motor Company. Boyles (1991, Journal of Quality Technology. 23: 17–26) and Spring (1995, Total Quality Management 6(3): 427–438.) point out that as soon as and S control charts are in statistical control, the control charts of process capability indices can be used to monitor the quality of process. In the previous, we know that if the process is not in control, the process capability index control chart can be used to monitor the differences of process capability, and as soon as the process is in control the stable process capability can be identified. Therefore, process capability index control chart not only can be used to monitor the stability of process’s quality but also can be used to monitor the quality of process. Since Boyles (1991, Journal of Quality Technology 23: 17–26.) and Spiring (1995, Total Quality Management 6(1): 21–33.) had had research about control chart of the bilateral specification index C _pm., but there are many kinds of products, which meet unilateral quality specification. Therefore, we will construct the control chart of unilateral specification index C _pl and C _pu to monitor and evaluate the stability of process and process capability.     

Process capability assessment for index C pk based on bayesian approach

Process capability indices have been proposed in the manufacturing industry to provide numerical measures on process reproduction capability, which are effective tools for quality assurance and guidance for process improvement. In process capability analysis, the usual practice for testing capability indices from sample data are based on traditional distribution frequency approach. Bayesian statistical techniques are an alternative to the frequency approach. Shiau, Chiang and Hung (1999) applied Bayesian method to index C_pm and the index C_pk but under the restriction that the process mean μ equals to the midpoint of the two specification limits, m. We note that this restriction is a rather impractical assumption for most factory applications, since in this case C_pk will reduce to C_p. In this paper, we consider testing the most popular capability index C_pk for general situation – no restriction on the process mean based on Bayesian approach. The results obtained are more general and practical for real applications. We derive the posterior probability, p, for which the process under investigation is capable and propose accordingly a Bayesian procedure for capability testing. To make this Bayesian procedure practical for in-plant applications, we tabulate the minimum values of Ĉ_pk for which the posterior probability p reaches desirable confidence levels with various pre-specified capability levels.     

Estimating process capability index C′′pmk for asymmetric tolerances: Distributional properties

Pearn et al. (1999) considered a capability index C ^′′ _pmk, a new generalization of C _pmk, for processes with asymmetric tolerances. In this paper, we provide a comparison between C ^′′ _pmk and other existing generalizations of C _pmk on the accuracy of measuring process performance for processes with asymmetric tolerances. We show that the new generalization C ^′′ _pmk is superior to other existing generalizations of C _pmk. Under the assumption of normality, we derive explicit forms of the cumulative distribution function and the probability density function of the estimated index . We show that the cumulative distribution function and the probability density function of the estimated index can be expressed in terms of a mixture of the chi-square distribution and the normal distribution. The explicit forms of the cumulative distribution function and the probability density function considerably simplify the complexity for analyzing the statistical properties of the estimated index . Received April 2000     

Yield measure for the process with multiple streams

Process capability indices, such as C _pk , have been widely used in the manufacturing industry to provide common quantitative measures for process performance. The index C _pk only provides an approximate rather than an exact measure of the process yield. To obtain an exact measure of the process yield, Boyles proposed a yield index S _pk . Capability measures for processes with single stream have been investigated extensively; however, multiple streams processes often occur in practice. Bothe presented a capability index for multiple streams process. In the present paper, a new index that is able to provide an exact measure of yield for a multiple streams process is developed. Three examples are given for illustration. From the results of the yield measure in the three examples, the conventional approach, using the arithmetic average of the estimated yield indices of all streams, will certainly over-estimate the process yield.     

Estimating and Testing Process Precision with Presence of Gauge Measurement Errors

Process capability indices have been widely used in the manufacturing industry. Those capability indices, quantifying process potential and performance, are important for any successful quality improvement activities and quality program implementation. Because of the simplicity and easy of understanding, the precision index C_p has gained its popularity for measuring process consistency. However, the quality of data on the process characteristics relies very much on the gauge measurement. Conclusions about capability of the process just only based on the single numerical value of the index are not reliable. In this paper, we not only conduct the performance of the index C_p with gauge measurement errors, but also present adjusted confidence interval bounds and critical values for capability testing purpose of C_p with unavoidable measurement errors. Our research would help practitioners to determine whether the factory processes meet the capability requirement, and make more reliable decisions.     

Measuring process capability index <Emphasis Type="Italic">C</Emphasis><Subscript><Emphasis Type="Italic">pm</Emphasis></Subscript> with fuzzy data

The process capability index C _pm , which considers the process variance and departure of the process mean from the target value, is important in the manufacturing industry to measure process potential and performance. This paper extends its applications to calculate the process capability index [(C)\tilde]_pm{\tilde {C}_{pm} } of fuzzy numbers. In this paper, the α-cuts of fuzzy observations are first derived based on various values of α. The membership function of fuzzy process capability index [(C)\tilde]_pm{\tilde {C}_{pm} } is then constructed based on the α-cuts of fuzzy observations. An example is presented to demonstrate how the fuzzy process capability index [(C)\tilde]_pm{\tilde {C}_{pm} } is interpreted. When the quality characteristic cannot be precisely determined, the proposed method provides the most possible value and spread of fuzzy process capability index [(C)\tilde]_pm{\tilde {C}_{pm} }. With crisp data, the proposed method reduces to the classical method of process capability index C _pm .     

Generalized confidence intervals for the process capability index C pm

Process capability indices have been proposed to the manufacturing industry for measuring process reproduction capability. The C _pm index takes into account the degree of process targeting (centering), which essentially measures process performance based on average process loss. To properly and accurately estimate the capability index, numerous conventional approaches have been proposed to obtain lower limits of the classical confidence intervals (CLCLs) for providing process capability information. In particular, lower confidence limits (LCLs) not only provide critical information regarding process performance but are used to determine if an improvement was made in reducing the nonconforming percent and the process expected loss. However, the conventional approach lacks for exact confidence intervals for C _pm involving unknown parameters which is a notable shortcoming. To remedy this, the method of generalized confidence intervals (GCIs) is proposed as an extension of classical confidence intervals (CCIs). For evaluating practical applications, two lower limits of generalized confidence intervals (GLCLs) for C _pm using generalized pivotal quantities (GPQs) are considered, (i) to assess the minimum performance of one manufacturing process/one supplier, and (ii) to assess the smallest performance of several manufacturing processes/several suppliers for equal as well as unequal process variances.     

Implementing lifetime performance index for the pareto lifetime businesses of the service industries

Process capability analysis is an effective means of measuring process performance and potential capability. In the service industries, process capability indices (PCIs) are utilized to assess whether business quality meets the required level. Hence, the performance index C _L is used as a means of measuring business performance, where L is the lower specification limit. In the technology of data transformation, this study constructs a uniformly minimum variance unbiased estimator (UMVUE) of C _L based on the right type II censored sample from the pareto distribution. The UMVUE of C _L is then utilized to develop a novel hypothesis testing procedure in the condition of known L. Finally, we give one practical example and the Monte Carlo simulation to assess the behavior of this test statistic for testing null hypothesis under given significance level. Moreover, the managers can then employ the new testing procedure to determine whether the business performance adheres to the required level.     

On the upper and lower semicontinuity of the Aumann integral

Let (T,τ,μ) be a finite measure space, X be a Banach space, P be a metric space and let L₁(μ,X) denote the space of equivalence classes of X-valued Bochner integrable functions on (T,τ,μ). We show that if φ:T×P→2^X is a set-valued function such that for each fixed pεP, φ(·,p) has a measurable graph and for each fixed tεT, φ(t,·) is either upper or lower semicontinuous then the Aumann integral of φ, i.e.,∫_Tφ(t,p)dμ(t)= {∫_Tx(t)dμ(t):xεS_φ(p)}, where S_φ(p)= {yεL₁(μ,X):y(t)εφ(t,p)μ−a.e.}, is either upper or lower semicontinuous in the variable p as well. Our results generalize those of Aumann (1965, 1976) who has considered the above problem for X=Rⁿ, and they have useful applications in general equilibrium and game theory.     

The non-existence of smooth demand in general banach spaces

We study the questions of existence and smoothness of demand functions with an infinite number of commodities. The main result obtained, under some hypothesis, is: if a C¹ demand exists in a commodity space B, then B can be given an inner product structure. For example, if B is L_p, 1p∞, and if there exists a C¹ demand function defined on B then p must be 2. Another result is: if a demand function exists, defined for all prices p and income, then the commodity space must be reflexive. For example, if B is L_p and a demand function exists on B, defined for all prices and incomes then 1<p<∞. We also study the cases L_∞ and L₁ with weaker assumptions. We finish the paper proving that the demand function is always defined for a dense set of prices and convenient incomes.     

Semiparametric estimation of a censored regression model with an unknown transformation of the dependent variable

This paper presents a method for estimating the model Λ(Y)=min(β′X+U, C), where Y is a scalar, Λ is an unknown increasing function, X is a vector of explanatory variables, β is a vector of unknown parameters, U has unknown cumulative distribution function F, and C is a censoring threshold. It is not assumed that Λ and F belong to known parametric families; they are estimated nonparametrically. This model includes many widely used models as special cases, including the proportional hazards model with unobserved heterogeneity. The paper develops n^1/2-consistent, asymptotically normal estimators of Λ and F. Estimators of β that are n^1/2-consistent and asymptotically normal already exist. The results of Monte Carlo experiments illustrate the finite-sample behavior of the estimators.     

A new process capability index

In this paper a new process capability index (C _pq ) has been introduced, which is easy to compute and performs well when compared with its natural competitor (C _pm ). Work done while he was Lukacs Visiting Professor on leave from the University of Maryland.     

Representing preferences with nontransitive indifference by a single real-valued function

Let be an interval order on a topological space (X, τ), and let x _˜^* y if and only if [y z x z], and x _˜^** y if and only if [z x z y]. Then _˜^* and _˜^** are complete preorders. In the particular case when is a semiorder, let x _˜⁰ y if and only if x _˜^* y and x _˜^** y. Then _˜⁰ is a complete preorder, too. We present sufficient conditions for the existence of continuous utility functions representing _˜^*, _˜^** and _˜⁰, by using the notion of strong separability of a preference relation, which was introduced by Chateauneuf (Journal of Mathematical Economics, 1987, 16, 139–146). Finally, we discuss the existence of a pair of continuous functions u, υ representing a strongly separable interval order on a measurable topological space (X, τ, μ, ).     

On a study of the exponential and Poisson characteristics of the Poisson process

Based on the exponential and Poisson characteristics of the Poisson process, in this work we present some characterizations of the Poisson process as a renewal process. More precisely, let γ_t be the residual life at time t of the renewal process A={A(t),t≥0 }, under suitable condition, we prove that if Var(γ_t)=E ² (γ_t),∀t≥0, then A is a Poisson process. Secondly, we show that if Var (A(t)) is proportional to E (A(t)), then A is a Poisson process also, and Var (A(t))=E (A(t)). Received: August 1999     

A method to obtain new copulas from a given one

Given a strictly increasing continuous function φ from [0, 1] to [0, 1] and its pseudo-inverse φ^[−1], conditions that φ must satisfy for C_φ(x₁, . . . ,x_n)=φ^[−1](C(φ(x₁), . . . ,φ(x_n))) to be a copula for any copula C are studied. Some basic properties of the copulas obtained in this way are analyzed and several examples of generator functions φ that can be used to construct copulas C_φ are presented. In this manner, a method to obtain from a given copula C a variety of new copulas is provided. This method generalizes that used to construct Archimedean copulas in which the original copula C is the product copula, and it is related with mixtures     

Characterization of the multivariate normal distribution by conditional normal distributions

Summary LetX andY be two random vectors with values in ℝ ^k and ℝ∝, respectively. IfZ=(X ^T,Y ^T) ^T is multivariate normal thenX givenY=y andY givenX=x are (multivariate) normal; the converse is wrong. In this paper simple additional conditions are stated such that the converse is true, too. Furthermore, the case is treated that the random vectorZ=(X ₁ ^T , …,X _t ^T ) ^T is splitted intot≥3 partsX ₁, …,X _t.     

Benchmark priors for Bayesian model averaging

In contrast to a posterior analysis given a particular sampling model, posterior model probabilities in the context of model uncertainty are typically rather sensitive to the specification of the prior. In particular, ‘diffuse’ priors on model-specific parameters can lead to quite unexpected consequences. Here we focus on the practically relevant situation where we need to entertain a (large) number of sampling models and we have (or wish to use) little or no subjective prior information. We aim at providing an ‘automatic’ or ‘benchmark’ prior structure that can be used in such cases. We focus on the normal linear regression model with uncertainty in the choice of regressors. We propose a partly non-informative prior structure related to a natural conjugate g-prior specification, where the amount of subjective information requested from the user is limited to the choice of a single scalar hyperparameter g_0j. The consequences of different choices for g_0j are examined. We investigate theoretical properties, such as consistency of the implied Bayesian procedure. Links with classical information criteria are provided. More importantly, we examine the finite sample implications of several choices of g_0j in a simulation study. The use of the MC³ algorithm of Madigan and York (Int. Stat. Rev. 63 (1995) 215), combined with efficient coding in Fortran, makes it feasible to conduct large simulations. In addition to posterior criteria, we shall also compare the predictive performance of different priors. A classic example concerning the economics of crime will also be provided and contrasted with results in the literature. The main findings of the paper will lead us to propose a ‘benchmark’ prior specification in a linear regression context with model uncertainty.     

Continua of stochastic dominance relations for bounded probability distributions

Let P={F,G,…} be the set of probability distribution functions on [0,b]. For each αε[1, ∞), F_αG means that ∫^x_o(x−y^α−1dF(y)∫^x_o(x−y)^α−1dG(y) for all xε[0, b], and F>_αG means that F_αG and F≠G. Each _α is reflexive and transitive and each>_α is asymmetric and transitive. Both _α and>_α increase as α increases but their limits are not complete. A class U_α of utility functions is defined to give F>_αG iff ∫udF>∫udG for all uεU_α. These classes decrease as α increases, and their limit is empty. Similar decreasing classes are defined for each _α, and their limit is essentially the constant functions on (0, b].