Process capability indices are quantitative measures on process performance. The C²p(u, v) indices family depending on two parameters allows to generalise the common indices used in industry. Ordinarily the process data and process requirements are assumed to be precise numbers. In practical situations, however, it is more realistic to assume that they are more or less imprecise. Such observations are called fuzzy. This paper presents a family of fuzzy indices for processes where the observations, the specification limits and the target are fuzzy. The membership function of fuzzy capability indices is constructed on the basis of the α-cuts of fuzzy data. A three-decision testing rule is developed to assess process performance based on p-values or critical values. With precise values the suggested approach boils down to the usual binary decision. A real example with a linguistic variable illustrates the setting up of the said method.

In the manufacturing industry, many product characteristics are of one-sided tolerances. The process capability indices Cpu (u, v) and Cpl (u, v) can be used to measure process performance. Most research work related to capability indices assumes no gauge measurement errors. This assumption insufficiently reflects real situations even when advanced measuring instruments are used. In this paper we show that using a critical value without taking into account these errors, severely underestimates the α-risk which causes a less accurate testing capacity. In order to improve the results we suggest the use of an adjusted critical value, and we give a Maple program to get it. An example in a polymer granulates manufactory is presented to illustrate this approach.

, For a process in which a shift of the mean from the target appears to be less serious in one direction than in the other, the user should be able to impose only one tolerance. If the risk is considered k times less serious in the direction opposite to the tolerance, Grau suggests using the C-p(u)(u, v) or C-p(l)(u, v) indices. In this paper we study the links between these indices and process centering, as well as the links between these indices and the percentage of non-conforming items manufactured under the assumption of normality. These results are used to choose the pair (u, v) meeting the needs of the user as well as possible. We also develop a decision making rule based on the natural estimator which can be used to test whether the process is capable or not. We then present a study made on a polymer granulates manufacturing process to illustrate how this reasoning can be applied.

Capability indices were introduced to allow comparing the performance of various processes independently of their tolerance interval. The concept of performance, related first to the proportion of conforming items (process yield), quickly evolved to also take into account the process position compared to its target (process centering). Many indices were thus suggested, and the families C p (u, v) in the case of symmetrical tolerances, then in the case of asymmetrical tolerances, allowed to generalize the most usual indices. If the links between capability indices and process centering were studied, those between capability indices and processes yield were studied only partially. In this article we clarify the links between the process yield and the indices for all the positive or null values of u and v.

The families of process capability indices C p ( u , v ) and C ” p ( u , v ) provide measurements of process performances for processes with symmetric or asymmetric tolerances. In literature, no attention has been paid to the cases in which sample data are affected by gauge measurement errors, except for the basic indices C p , C pk , C pm and C pmk . However, these errors are always present in real situations even when advanced measuring instruments are used. If these errors are not taken into account, conclusions drawn from process capability are therefore unreliable. In this paper, we study the probability distribution of the estimator of C ” p ( u , v ) when the observations are affected by gauge measurement errors. We show here that using a lower confidence bound without taking these errors into account, severely underestimates the true capability. In order to improve the results, we suggest using an adjusted lower confidence bound, and we give a Maple program to obtain this bound. We finally present a real study on a carbon fibre manufacturing process to illustrate how to make use of our suggestion.

Most research works related to process capability indices assume no gauge measurement errors. However, such an assumption inadequately reflects real situations even when advanced measuring instruments are employed. If we do not take into account these errors, conclusions drawn from process capability are therefore unreliable. In this paper we study the sampling distribution of capability indices Cp''(u,v) in the presence of measurements errors, and when small subsamples data are collected from past "in-control". We show that using a critical value without taking into account these errors, severely underestimates the α-risk which causes a less accurate testing capacity. To improve the results we suggest the use of an adjusted critical value, and we give a Maple program to get it. An example in a nougat manufactory is presented to illustrate this approach.

Les indices de capabilite ont ete introduits pour mesurer la performance d'un processus de production industrielle. Le concept de performance, initialement lie a la proportion de pieces non conformes fabriquees, a rapidement evolue pour tenir compte aussi de la position de la moyenne du processus par rapport a sa cible. Si les liens entre les indices et le centrage ont deja ete etudies, ceux entre les indices et la proportion de non conformes ne l'ont ete que partiellement. Dans cet expose nous clarifions ces liens et montrons sur un exemple reel comment ces resultats peuvent etre utilises.

When the distribution of one of the characteristics of a process is non normal, methods based on empirical percentiles suggest the use of several process capability indices (PCIs) which are similar to the usual Cp, Cpk, Cpm, and Cpmk indices. However most of these PCIs apply only to the case of symmetrical tolerances. To take into account the asymmetry of the tolerances as well as the asymmetry of the process distribution, new PCIs which improve the previous ones are proposed. In the end and in order to validate the method proposed here, we apply it to a real production case.

Process capability indices Cp, Cpk, Cpm and Cpmk have been proposed to the manufacture industry to provide measures of process performance. The family C″p(u, v) depending on two parameters allows to generalise the previous indices for any position of the target in the tolerance interval. Grau (2010) studied the links between the value of the C″p(u, v) index and the maximum deviation of the process mean, as well as the maximum percentage of non-conforming items manufactured under the assumption of normality. From these results, we provide tables which allow the practitioner to choose a pair (u, v), so that the resulting index C″p(u, v) will meet his objectives best. We develop a decision making rule based on the natural estimator which can be used to test whether the process is capable or not. We then present a real study made on a carbon fibre manufacturing process to illustrate how this reasoning can be applied.

For a process normally distributed and a target located at the midpoint of a two-sided tolerance interval, Vannman [10] suggests a family of indices depending on two parameters. In order to choose the parameters for obtaining an index that is sensitive to departures of the mean from the target, Vannman takes the properties of its estimator into account. For a target not located at the midpoint of the tolerance interval, Chen and Pearn [2] suggest a family of indices generalizing Vannman's. The moments of the estimators of these indices are studied to choose an index whose the estimator is sensitive to departures of the mean from the target, and has a small bias, and a small mean square error.

The unbalanced non-central chi-square distribution with 1 degree of freedom, introduced (and called weighted non-central chi-square distribution) by Chen is generalized to the case of ν degrees of freedom. Thus we obtain the non-central moments as well as the central moments in specific cases.

Capability indices are dimensionless quantities measuring the ability of a process to manufacture items whose characteristics must be within a specified tolerance range. In this case and for a normally distributed process, the indices C(p), C(pk), C(pm), and C(pmk) are widely used. In this paper we study the case where a single tolerance is imposed because the shifts in the direction of this tolerance seem more serious than in the opposite direction. We propose a family of four indices having better properties than the existing indices. These new indices are created from the usual properties and interpretations of the indices C(p), C(pk), C(pm), and C(pmk), and can be used without difficulty in industry.

Capability indices are dimensionless quantities measuring the aptitude of a process to manufacture items whose characteristics must be within a specified tolerance ranges. The usual indices $C_p$, $C_{pk}$, $C_{pm}$, and $C_{pmk}$ are used for a process of normal distribution and a target located at the center of the tolerance interval. Various indices derived from the previous family allow to consider more complex situations when asymmetrical tolerances and non-normal distributions are taken into account. In this paper we study the case where a single tolerance is imposed because the shifts in the direction of this tolerance appear much more serious than in the opposite direction. We propose a family of four indices having interpretations and properties similar to those of the usual family.

The capability indices for only one characteristic of quality with symmetrical tolerances have been widely studied. This paper deals with the situation of several characteristics supervised simultaneously, for a rectangular tolerance region and an unspecified target inside this region. New capability indices are suggested, which generalize Chen’s, Lin’s, and Pearn’s indices C″p, C″pk, C″pm, and C″mk, defined for the case of only one variable with asymmetric tolerances, as well as the C_p, C_pk, C_pm, and C_pmk indices, defined by Grau for the case of several variables with symmetrical tolerances.

The weighted non-central chi-square distribution with 1 degree of freedom, introduced by Chen is generalized to the case of $\nu$ degrees of freedom. Thus we obtain the non central moments as well as the central moments in specific cases.

ABSTRACT There have been many investigations on the capability indices C p , C pk , C pm , and C pmk , for the common situation in which the target is the midpoint of the tolerance interval. However, only a few investigations deal with the specific case of asymmetrical tolerances. In that particular case, a number of symmetrical and asymmetrical indices are put forward, but there is no full literature treatment or synthesis showing the similarity between those indices and the common ones. We intend here, to demonstrate that the algebraic links between the indices C p , C pk , C pm , and C pmk , are similar to the ones which relate the symmetrical indices proposed in the case of asymmetrical tolerances. In that case, the algebraic structure allows us to propose asymmetrical indices families. An example based on a pharmaceutical filling operation is used to illustrate the application.

Ecophysiological differences related to photosynthesis were compared in holm oak Quercus ilex leaves from undisturbed holm-oak vegetation, resprouts after fire and resprouts after tree-fell. No significant differences in any parameter measured were observed between the two kinds of resprout throughout the first growing season following disturbance. Resprouting leaves showed lower carbon isotope discrimination (Δ) and intercellular CO2 concentration (pi), and higher photosynthesis, leaf conductance and transpiration rates than leaves from undisturbed stands. Nitrogen, soluble protein content and ribulose bisphosphate carboxylase (RuBPCase) activity were 88%, 96% and 45% higher respectively, in both kinds of resprout. The results indicate that photosynthetic capacity, rather than stomatal conductance, is the limiting factor in photosynthesis in resprouts, Chlorophyll content and chlorophyll a/b ratio did not differ between resprouts and undisturbed leaves, indicating that the observed differences were not a result of differences in light environment during leaf development. Leaf mass per area (LMA), was 80% higher in the resprouts, and was negatively related (r=−0.86) to Δ and positively related (r=0.87) to N content. Enhanced carbon assimilation after disturbances resulted in higher water use efficiency, as indicated by lower Δ values in the resprouts. We conclude that the cause of defoliation was not relevant in the physiology of the resprouts, suggesting the importance of underground organs.