C pm MPPAC for manufacturing quality control applied to precision voltage reference process

A multiprocess performance analysis chart (MPPAC), based on the process capability index C _pm , called C _pm MPPAC, is developed to analyse the manufacturing quality of a group of processes in a multiple process environment. The C _pm MPPAC conveys critical information about multiple processes regarding the departure of the process and process variability on one single chart. Existing research on MPPAC has been restricted to obtaining quality information from one single sample of each process, ignoring sampling errors. The information provided from the existing MPPAC chart, therefore, is unreliable and misleading, resulting in incorrect decisions. In this paper, the natural estimator of C _pm is considered based on multiple samples. Based on the natural estimator of C _pm , sampling errors are considered by providing an explicit formula with Matlab to obtain the estimation accuracy of the C _pm . The sampling accuracy of C _pm is tablulated for sample size determination so that engineers/practitioners can use it for in-plant applications. An example of multiple PVR processes is presented to illustrate the applicability of C _pm MPPAC for manufacturing quality control.     

Economical operation of the C pm control chart for monitoring process capability index

Process capability indices are widely used to provide the evaluation measure of a process. Especially, the process capability index C _pm , which is defined by the range of the process standard specification limits and the deviation from a target value, is called the Taguchi index. Boyles has investigated the statistical characteristics of the estimator $\hat{C}_{pm}$ , and also proposed a technique for the C _pm control chart. Since the process capability index C _pm is based on the concept of the Taguchi’s quality loss, the process capability index C _pm already includes an economical concept. In this article, we evaluate an operating cost consisting of the sampling cost, the sample cost, and the quality loss of failing to detect an out-of-control state when the C _pm control chart is used. Then, we derive an optimal operating plan by sample size and sampling interval in order to minimize the ceiling value of the operating cost based on the min–max criterion.     

A review and extensions of sample size determination for estimating process precision and loss with a designated accuracy ratio

Numerous process capability indices, including C_p, C_pk, C_pm, and C_pmk, have been proposed in the manufacturing industry providing numerical measures of process capability based on various criteria. The index C_p provides a measure of process precision, which reflects the consistency of product quality. The index C_pm, also called the Taguchi index, essentially measures process loss. Lower confidence bounds estimate the minimum process capability conveying critical information regarding product quality, which is essential to quality assurance. Existing research works have focused on constructing lower confidence bounds, but investigation on the sample sizes required for a specified accuracy ratio of the estimation has been comparatively neglected. The sample size determination is important, as it is directly related to the cost of the data collection plan. In this paper, we review some existing formulas for lower confidence bounds that can be used to determine the sample size required for a given estimating accuracy ratio. We then present a different approach, with efficient MATLAB programs, to obtain the sample sizes using the UMVUE (uniformly minimum variance unbiased estimator) of C_p, and the MLE (maximum likelihood estimator) of C_pm. We also provide tables of the sample size information for the engineers/practitioners to use for their in-plant applications.     

Measuring manufacturing capability based on lower confidence bounds of C pmk applied to current transmitter process

Several process capability indices, including C_p, C_pk, and C_pm, have been proposed to provide numerical measures on manufacturing potential and actual performance. Combining the advantages of those indices, a more advanced index C_pmk is proposed, taking the process variation, centre of the specification tolerance, and the proximity to the target value into account, which has been shown to be a useful capability index for manufacturing processes with two-sided specification limits. In this paper, we consider the estimation of C_pmk, and we develop an efficient algorithm to compute the lower confidence bounds on C_pmk based on the estimation, which presents a measure on the minimum manufacturing capability of the process based on the sample data. We also provide tables for practitioners to use in measuring their processes. A real-world example of current transmitters taken from a microelectronics device manufacturing process is investigated to illustrate the applicability of the proposed approach. Our implementation of the existing statistical theory for manufacturing capability assessment bridges the gap between the theoretical development and the in-plant applications.     

Capability measure for asymmetric tolerance non-normal processes applied to speaker driver manufacturing

Process capability indices, C_p(u,v), including C_p, C_pk, C_pm, and C_pmk, have been proposed in the manufacturing industry to provide numerical measures on process potential and performance for normal processes. Earlier studies considered a class of flexible capability indices, called C_Np(u,v), for processes with non-normal distributions where the tolerances are symmetric. In this paper we consider an extension of C_Np(u,v), called C”_Np(u,v), to handle non-normal processes with asymmetric tolerances. The extension takes into account the important property of the asymmetric loss function, which is shown to be more sensitive to process shift and more accurate than C_Np(u,v) in measuring process capability, hence provides better manufacturing quality assurance. Comparisons between C_Np(u,v) and the extension C”_Np(u,v) are provided. We propose a sample percentile estimator, and apply the bootstrap method to find the lower confidence bound for testing manufacturing capability. We also develop an integrated S-PLUS program to calculate the percentile estimator and the corresponding lower confidence bound. As an illustration, the proposed approach is applied to capability testing of home-theater speaker systems.     

Testing manufacturing performance based on capability index C pm

The loss-based process capability index C_pm, sometimes called the Taguchi index, has been proposed to the manufacturing industry to measure process performance. In this paper, we propose a method to assess the performance of a normally distributed process. We implement the theory of testing hypothesis using the natural estimator of C_pm, and provide an efficient program to calculate the p-values. We also provide tables of the critical values for some commonly used capability requirements. Based on the test, we develop a simple step-by-step procedure for in-plant applications.     

Selecting a supplier by fuzzy evaluation of capability indices C pm

A process capability index C_pm that fits nominal-the-best type quality characteristics is an effective tool for assessing process capability since the index can adequately reflect a centring process capability and process yield. A valuable method using C_p estimators was developed by Chou [7] for practitioners to use to determine whether or not two processes have equal capability. However index C_p failed to measure process yield and process centring with bilateral specifications and in addition, more than two suppliers can be selected in an actual application. This study proposes a fuzzy inference method to select the best among the competing suppliers based on an estimated capability index of C_pm calculated from sampled data. This method has the advantages of fuzzy systems where a grade can be obtained instead of a more specific exact evaluation result. An illustrated example of colour STN displays demonstrates that the proposed method is effective and feasible for the evaluation of competing process capability.     

Procedures for testing manufacturing precision Cpbased on (\bar{x},R) or (\bar{x},S) control chart samples

Process precision index C_p has been widely used in the manufacturing industry for measuring process potential and precision. Estimating and testing process precision based on one single sample have been investigated extensively. In this paper, we consider the problem of estimating and testing process precision based on multiple samples taken from ( ,R)or ( ,S)control chart. We first investigate the statistical properties of the natural estimator of C_p and implement the hypothesis testing procedure. We then develop efficient MAPLE programs to calculate the lower confidence bounds, critical values, and p-values based on m samples of size n. Based on the test, we develop a step-by-step procedure for practitioners to use in determining whether their manufacturing processes are capable of reproducing products satisfying the preset precision requirement.     

Measuring process yield based on the capability index Cpm

Process capability indices C_p, C_a, C_pk and C_pm have been proposed to the manufacturing industry as capability measures based on various criteria including variation, departure, yield, and loss. It has been noted in recent quality research and capability analysis literature that both the C_pk and C_pm indices provide the same lower bounds on process yield, that is, Yield2(3C_pk)-1=2(3C_pm)-1. In this paper, we investigate the behaviour of the actual process yield in terms of the number of nonconformities (in ppm) for processes with a fixed index value of C_pk=C_pm, but with different degrees of process centring, which can be expressed as a function of the capability index C_a. The results illustrate that it is advantageous to use the index C_pm over the index C_pk when measuring process capability, since C_pm provides better customer protection. This revised version was published online in October 2004 with a correction to the issue number.     

The evaluation of process capability for a machining center

The machining center methodology is widely applied to production systems within the fast-developing processing industry. The automated machining center can simultaneously perform certain processes, such as milling, drilling, boring, and reaming, on the same machine at the same time, and manufactures limited quantities of multiple-type products. These products usually have numerically important quality characteristics with high accuracy. However, a machining center is unable to measure process capability on its own. Thus, we reflect the process capability of a machining center by measuring the quality characteristics of its processing products. In this paper, we propose the three integrated process capability indices (PCIs) C _p^mc, C _a^mc, and C _pm^mc to evaluate the integrated process precision, accuracy, and actual capability respectively. Furthermore, we develop a process capability monitoring figure (PCMF), which not only displays the status of the process precision and accuracy by the color management method [1], but it also forecasts the integrated process capability of the next productive time (batch) through the analysis of time series. Using the PCMF, engineers will be assisted with tasks such as monitoring the process quality, deciding the period of the borer’s replacement, and designing the process parameters.     

Evaluating process centering with large samples – an approximate lower bound on the accuracy index

Process accuracy index C_a has been proposed in the manufacturing industry to provide numerical measures on process centering (the ability to cluster around the specification mid-point). Investigations on process accuracy in existing engineering statistics and quality assurance literature are all focused on small samples with a normality assumption. In this paper, a natural estimator of process accuracy index is considered. The limiting distribution and related large sample properties of the considered estimator are studied under general populations having fourth central moment exists. A decision-making procedure based on an approximate 100(1-α)% lower bound of the process accuracy index is also constructed. A practical example is demonstrated to illustrate how the proposed procedure may be applied for in-plant applications to judge whether the process runs under the desirable accuracy requirement.     

Comparison between two process capability indices using generalized confidence intervals

Processcapability indices (PCIs) are extensively used statistical measures to assess process performance in manufacturing industry. In this paper, confidence intervals for the difference between PCIs for two processes are derived by the generalized pivotal quantity method. The indices C _pk , C _pmk , and C _pm are considered in this study. The performance of the proposed method is assessed using simulation study. The results are also illustrated using an example from industrial contexts.     

A random interval estimation of the estimated process accuracy index

Process accuracy index C_a has been proposed in the manufacturing industry to provide a numerical measure to assess process performance with respect to accuracy. Information regarding process centering (the ability to cluster around the specification mid-point) in existing engineering statistics and quality assurance literature is relatively rare. In this paper, a natural estimator of the accuracy index is considered under the normality assumption. The exact probability density function and the rth moment raised from a non-central chi-squared distribution of the estimated process accuracy index are derived. A decision-making procedure based on a random interval estimation of the considered index is also constructed. A practical example is demonstrated to illustrate how the proposed reliable procedure may be applied for in-plant applications.     

Evaluation model for multi-process capabilities of stranded wire

Stranded wire is the most important component of familiar mechanical equipment such as elevators, cable cars, and cranes. The quality of these products that are used on a daily basis are mainly affected by the tensile strength of stranded wire. In order to attain the purpose of economical design and a long life span of stranded wire, a less relaxation property of strand type is suitable for manufactured tools. Thus, the manufacturing industries of stranded wire need to reach the goals of high tensile strength and low relaxation. To ensure the required quality of stranded wire, the strand pull test and the long period relaxation test are two important quality assurance tests. There are three specific items of the tensile strength test that belong to the larger-the-better quality type. The quality type of the smaller-the-better is for the long period relaxation test. However, many existing methods are able to measure process capability for the product with a single quality characteristic although it cannot be applied to most products with multiple properties. Thus, the indices of C_pu and C_pl, for the larger-the-better and the smaller-the-better quality type respectively proposed by Kane [5], are quoted and combined to propose a new index to evaluate the quality of multiple characteristics of stranded wire in this article. The principle of statistics is then used to derive the one-to-one mathematical relationship of this new index and ratio of satisfactory production process. Finally, the procedure and criteria to evaluate the quality of stranded wire is proposed. This integrated multi-quality property capability analysis model can be used to evaluate the multi-process capabilities and provide continuous improvements on the manufacturing process of stranded wire.     

Process capability analysis based on minimum production cost and quality loss

Conventional process capability indices, such as C _p, C _pk, and C _pm, have been used widely and successfully in most of today’s manufacturing sectors. These process capability indices, however, can only measure the process consequence resulting from various combinations of process mean and process variance on site by measuring the output without concern as to cost. Hence, the focus of this research was to develop a new process capability index, C _pmc, which not only considered the impact from process mean and process variance, as is conventionally done, but also considered the relevant quality and production costs incurred in a product’s life cycle. With these added aspects, the new process capability index, C _pmc, is considered as a response value which needs to be optimized to ensure that both an economical and a quality design can be achieved. The design constraints are the restrictions resulting from the process capability limits, the feasible process range, and the functionality requirements. An example of product assembly is introduced to demonstrate the applicability of the presented approach so that the optimal mean and tolerance values can be determined and so that the C _pmc value is maximized. By using the presented C _pmc, a high-quality but low-cost process design can be achieved during the early stages of product design and process planning.     

Contract manufacturer selection by using the process incapability index C pp

In the competitive global business environment, enterprises should effectively respond to and satisfy customer needs. Outsourcing manufacturing is an effective method of adjusting the flexibility of production capability. To ensure final product quality, evaluating and selecting a contract manufacturer is essential. Process capability indices are extensively adopted in manufacturing to determine whether a process meets capability requirements. This study attempts to determine the score index R_i and then apply it to assess the process performance of contract manufacturer by using the process incapability index C _pp . The formulae for C _pp and R_i are simple to understand and to apply. The procedure developed in this paper is also an easy and convenient tool for practitioners to assess contract manufacturer quality performance and make more reliable decisions regarding contract manufacturer.     

Optimal concurrent tolerance based on the grey optimal approach

A new method for optimal concurrent process tolerance is proposed. Information related to process planning is used during the structure design stage of a product. The functional tolerances of the assembly of a product are considered process tolerances. A nonlinear optimal concurrent process tolerance model has been established to minimize the total manufacturing cost with different weight factors for the operations of the product. The constraints include concurrent process tolerance chains, the standard coefficient of process tolerance, and the economical maximum tolerance of the machine tools. In order to obtain optimal process tolerances, a new approach based on grey difference degree is presented in this paper. This approach takes the sequence consisting of the optimal value of every objective as the standard sequence, and that consisting of the actual values as the objective sequence. Thus, grey difference degree is calculated and used as the objective of the function. Finally, a practical example is introduced to demonstrate the effectiveness of the proposed method.Nomenclature Y ₀ standard sequence - Y ₀(k) k-th component of the standard sequence - Y _i objective sequence - Y _i (k) k-th component of the objective sequences - (k) the grey correlation coefficient of Y _i relative toY ₀ at point k - y _i the i-th required tolerance specification of assembly structure - TX design tolerance vector of assembly structure - r _u the total number of relative process steps for u-th part - T _uv v-th relative process tolerance vector of u-th part - _uv selection coefficient of process sequence - w _uv the weight coefficient of the process tolerance T _uv - C _uv manufacturing cost of the process tolerance T _uv - C _uv ^* the ideal value of C _uv - z _uv the standard normal transform coefficient relative to process tolerance T _uv - _uv the standard deviation of T _uv      

Adaptive sampling interval cause-selecting control charts

This article considers the statistical adaptive process control for two dependent process steps. We construct an adaptive sampling interval Z _X control chart to monitor the quality variable produced by the first process step, and use the adaptive sampling interval Z _e control chart to monitor the specific quality variable produced by the second process step. By using the proposed adaptive sampling interval control charts, we can quickly detect and distinguish which process step is out of control. The performance of the proposed adaptive sampling interval control charts is measured by the adjusted average time to signal (AATS), which was derived by a Markov chain approach, for an out-of-control process. An empirical automobile braking system example shows the application and the performance of the proposed adaptive sampling control charts in detecting shifts in process means. Some numerical results obtained demonstrated that the performance of the proposed adaptive sampling cause-selecting control charts outperforms the fixed sampling interval cause-selecting control charts.     

Bayesian approach for measuring EEPROM process capability based on the one-sided indices CPU and CPL

The purpose of process capability analysis is to provide numerical measures on whether a process is capable of reproducing items meeting the manufacturing specifications. Capability analyses have received considerable recent research attention and increased usage in process assessments and purchasing decisions. Most existing research works on capability analysis focus on estimating and testing process capability based on the traditional distribution frequency approach. In this paper, we propose a Bayesian approach based on the indices C_PU and C_PL to measure EEPROM process capability, in which the specifications are one-sided rather than two-sided. We obtain the credible intervals of C_PU and C_PL and develop a Bayesian procedure for capability testing. The posterior probability p, for which the process under investigation is capable, is derived. The credible interval is a Bayesian analog of the classical lower confidence interval. A process satisfies the manufacturing capability requirements if all the points in the credible interval are greater than the pre-specified capability level w. To make this Bayesian procedure practical for in-plant applications, a real example of an EEPROM manufacturing process is investigated, demonstrating how the Bayesian procedure can be applied to actual data collected in the factories.     

A methodology for assessing manufacturing cost due to tolerance of aerodynamic surface features on turbofan nacelles

A component of the direct operating cost of aircraft is that associated with the manufacturing cost. This affects depreciation, interest, insurance and maintenance charges. By relaxing the requirements for aerodynamic surface smoothness the manufacturing cost can be reduced at the expense of an increase in drag and corresponding fuel costs. This work is part of a study to examine this multidisciplinary problem. Only isolated turbofan nacelles are considered. The costs associated with assigning different tolerance levels to the feature dimensions on nacelles are assessed. A statistical procedure is employed to estimate the cost-tolerance relationship for eleven features involving gaps, steps, surface profile and fastener flushness. This procedure requires actual manufacturing and cost source data. A knowledge of the cost-tolerance relationships is useful in a concurrent engineering context. It will allow aerodynamicists to optimise surface smoothness in consultation with production engineers, thus achieving the best compromise between cost and drag.Nomenclature A ₁ statistic distribution area to the left ofL ₁ - A _u statistic distribution area to the right ofL _u - C manufacturing cost (£/piece) - C _a assembly process manufacturing cost (£/piece) - C _c cost per concession (£/concession) - C _ci interface component edgei manufacturing cost (£/piece) - C _i interface manufacturing cost (£/piece) - C _lmh labour man hour cost (£/piece) - C _ma management cost (£/piece) - C _mh man hour cost (£/h) - C _mmh management man hour cost (£/piece) - C _nr non-recurrent cost (£/piece) (cost of all the jigs and tools used in the assembly process divided by the number of units produced) - C _rd redeployment cost (£/piece) - C _smh skilled worker man hour cost (£/h) - C _su support cost (£/piece) - C _sumh support man hour cost (£/h) - C _usmh unskilled worker man hour cost (£/h) - C ₀ initial manufacturing cost (£/piece) - d _tma t _ma parameter - d _{N c} N _c parameter - d _{O f} O _f parameter - d _{t r} t _r parameter - d _{t rd} t _rd parameter - d _{t su} t _su parameter - E _i interface edgei - F _i featurei - G interface gap (mm) - G _n interface gap normal to air flow (mm) - G _p interface gap parallel to air flow (mm) - L _ef production line loss of efficiency factor due to redeployment - L ₁ dimension lower specified limit (mm) - L _u dimension upper specified limit (mm) - M mean - N dimension nominal value (mm) - N _c number of concessions (concession/piece) - O _f worker overtime fraction - P surface profile deviation (mm) - R _on ratio between the worker overtime and the normal man hour cost - S interface step or fastener flushness (mm) - standard deviation - S _n interface step normal to air flow (mm) - S _p interface step parallel to air flow (mm) - T manufacturing tolerance (mm) - T _a assembly process manufacturing tolerance (mm) - t _bw basic work time (hour/piece) - T _ci interface component edgei manuacturing tolerance (mm) - T _i interface tolerance (mm) - t _ma management time (hour/piece) - T _p performance tolerance - t _r rework time (h/piece) - t _rd worker redeployment time to other production line stations different to the usual one (h/piece) - t _su support time of the methods, quality and design departments to the line (h/piece) - T _t tolerance threshold between skilled and unskilled worker employment (mm) - T ₀ initial manufacturing tolerance (mm)