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Abstract

Quantifying the current and expected future performance of a machining process no doubt is essential for continuous improvement of product quality and productivity. However, process capability evaluation for small batch production runs is a challenging work, because the assumptions required by traditional evaluation approaches based on statistical process control techniques are commonly not satisfied in real world. In this article, a sensitivity analysis–based process capability evaluation method for small batch production runs is proposed, and a new capability index is also presented correspondingly. In this method, an error propagation model between machining errors and input errors is first established using weighted least squares support vector machine. Then, the sensitivity distribution of machining errors versus input errors is characterized by a set of eigenvalues and eigenvectors in the variation space of input errors. Third, the safe variation space of input errors is solved according to the specification limits of quality characteristics. Finally, the process capability is evaluated by comparing the fitness between the safe variation space and the tolerance space of input errors. A practical case is addressed to validate the feasibility and effectiveness of the proposed method, and the results demonstrate that the method can measure a real small batch machining process effectively and get rid of the common assumption of independent and identical distribution, which is needed by traditional methods.

Keywords

Process capability small batch production sensitivity analysis weighted least squares support vector machine

Introduction

In today’s global competitive marketplace, adequate and reliable product quality is one of the critical factors of gaining the market competition advantages for manufacturing companies.^1,2 Quantifying the current and expected future performance of a machining process no doubt is essential for continuous improvement of product quality and productivity. Process capability indices (PCIs) that have been widely used in manufacturing industries and service industries provide numerical measures on whether or not a manufacturing process is capable to meet the predetermined specification level of a product.³ And the common used PCIs, such as $C_{p}$ , $C_{pk}$ , $C_{a}$ , $C_{pm}$ and $C_{pmk}$ , form a complementary system of performance measure for a machining process.⁴ These PCIs are based on the following assumptions:⁵ (A1) the interest quality characteristics are independent and identically normally distributed (IIND); (A2) the processes are in the state of statistical control and (A3) plenty of samples for estimating parameters are available. With tremendous growth of mass customization and extensive use of just-in-time techniques, small batch production runs have been popular in most discrete manufacturing companies, especially in the companies producing high-value cutting-hard parts, such as the key parts of aircraft landing gears and gas turbines. The machining processes of these high-value parts are characterized by small batch size, short lead time and little available quality data. Because of expensive blank and manufacturing cost, they need 100% inspection policy in every key stage to reduce quality accident to the minimum, which results that the quality observations are likely to be auto-correlated. On the other hand, a job (lot) may be split into a number of smaller jobs (sub-lots) using lot streaming (LS) technique so that successive operations of the same job can be overlapped to shorten the completion time of the whole job.^6–8 In such a scenario, the lot size of production may be very small. The small batch machining processes cannot be ensured in statistical control state due to short lead time and lack of sample data, although the machine error may fall in the specification limits. We call this non-statistical control but capable state as “dynamically stability sate.” Obviously, the traditional methods of process capability analysis cannot be quite qualified in such a case. Consequently, the process capability evaluation for small batch production runs, especially in the dynamically stable machining processes for high-value parts, is a serious and urgent challenge for quality engineers.

Focusing on the problem of process performance analysis for small batch production runs, a novel sensitivity analysis–based method for dynamic process capability evaluation (DPCE) is proposed in this article, and a new PCI is also presented. It is expected to establish a more practical measuring method for small batch machining processes which are usually in dynamically stable state. Because there is no unambiguous definition of small batch production, it is necessary to clarify the connotation of small batch production investigated in this study. In the implementation of control charts, 25 or more subgroups which consist of at least 100 observations need to be used to establish reliable control limits. Quessenberry⁹ suggested that 300 observation samples are used to establish the control limits of X chart. From the perspective of statistical quality control, when the available quality samples are not enough to implement Shewhart control charts, the production run can be called as small batch production. The purpose of this work is to analyze the process capability for this case.

The rest of this article is organized as follows. After a brief review about process capability evaluation for small batch machining processes in section “Literature review,” the logic framework of the proposed DPCE method is put forward in section “Logic framework of DPCE based on sensitivity analysis.” Then, the methodologies for construction of machining error propagation model based on weighted least squares support vector machine (WLS-SVM), sensitivity analysis of machining errors and evaluation of process capability are stated in sections “Construction of machining error propagation model based on WLS-SVM,”“Sensitivity analysis of machining errors” and “Evaluation of process capability based on sensitivity distribution” respectively. In section “Case study and discussions,” a practical case is studied to demonstrate the feasibility and effectiveness of the proposed method, and some discussions are also given. Finally, conclusions and future work are drawn in section “Conclusion.”

Literature review

As for the issue of process capability evaluation for small batch production runs, some important work has been done from the late 1980s. In the classic approaches based on statistical process control (SPC) techniques, it is a critical problem to accurately estimate the process parameters using available few samples. Aiming at the problem of small sample size, Cheng and Spiring,¹⁰ Saxena and Singh¹¹ and Miao et al.¹² adopted Bayesian approach to estimate the process parameters. Balamurali¹³ developed new appropriate capability indices on the basis of classic PCIs and further used bootstrap technique to estimate their confidence intervals. Ke et al.¹⁴ employed six bootstrap methods to construct upper confidence bounds of incapability index $C_{pp}$ for short-run production processes. Little and Harrelson¹⁵ presented an alternative method for conducting capability studies on limited data sets using t-distribution. Kounis et al.¹⁶ investigated how to calculate capability indices using extreme value statistics and discussed the potential for this method to replace existing methods. Satyala and Pieper¹⁷ presented a unified approach for predicting long-and short-term capability indices with dependence on machining target bias. Wang¹⁸ developed a procedure for constructing multivariate PCI for short-run production using the technique of principal component analysis. In addition, some researchers employed small sample process capability techniques for process monitoring¹⁹ and selection of the best process.²⁰ Although these capability analysis methods for small batch production remit the problem of unavailable large samples size to some extent, they are still under the assumption of IIND and statistical control. In the practical machining processes for high-value cutting-hard workpiece, there are no chances to collecting enough samples to analyze and eliminate assignable causes. Therefore, the assumption of IIND and statistical control is obviously not tenable, which results that the estimators of distribution parameters may be bias. Hence, the evaluation of PCIs may be incorrect. Gunter²¹ and English and Taylor²² indicated that the use of conventional PCIs may lead to overestimates of process quality and result in erroneous engineering decisions.

For achieving a consistent performance of a process, Taguchi introduced the concept of process robustness in manufacturing, which facilitate the application of robust design concept in engineering processes and products.²³ The existing approaches to measure the robustness and performance of a process or product can be classified into three categories, namely index-based approaches, experimental-based approaches and sensitivity-based approaches.²⁴ Among these approaches, most of index-based approaches are based on the concept of variance. In the most popular experimental-based approach, namely response surface methodology (RSM), it is assumed that the errors follow normal distribution and the error variance is constant and independent of mean. Hence, the two kinds of approaches are not suitable for small batch production.²⁵ To study robustness using sensitivity-based approaches, it is essential to construct an appropriate response model, which can avoid the assumption of IIND. Consequently, this kind of methodologies may be promising approaches to solve the issue of DPCE for small batch production. In the last decade, the trend of the study is to apply sensitivity-based approaches in measuring the robustness of a process or a product.²⁶ And sensitivity analysis–based methodologies have been used to solve the problems of performance or robustness evaluation in design and manufacturing fields. Zhu and Ting,²⁷ and Caro et al.²⁸ proposed a novel robust design technique based on the performance sensitivity distribution of a mechanism. Gunawan and Azarm²⁹ present a robust design optimization method based on the notion of a sensitivity region, which is a measure of how far a feasible design is from the boundary of a feasible domain in the parameter variation space. Dai and Scott³⁰ used sensitivity analysis and cluster analysis to improve both efficiency and effectiveness of a scale-based product family. In manufacturing fields, Zhang et al.³¹ used sensitivity analysis approach to evaluate the impact of each process-level error on the product quality, so as to make the optimization of fixture design as well as workpiece in-process tolerance allocation. Abellan-Nebot et al.³² employed the sensitivity analysis of candidate process plans to identify critical fixtures and manufacturing stations or operations. In the works mentioned above, the sensitivity analysis–based methods are used to define evaluation criteria, in order to optimize a design, select an optimal design from several different schemes, perform workpiece in-process tolerance allocation or identify critical fixtures and manufacturing stations. However, the nature of process capability evaluation is to compare the fitness between process performance and the predetermined specification limits. Therefore, these methodologies for robust design and process optimization cannot be applied in capability evaluation for machining processes. To measure the robustness of multi-stage manufacturing processes, Mondal et al.³³ adopted sensitivity matrices (Jacobian matrices) to construct the variation models in single-stage process. Nevertheless, their method is only appropriate for large batch production, because plenty of samples are required to estimate the standard deviation of controllable input variables. Even so, these applications demonstrate that sensitivity analysis–based method has a promising potential in measuring the performance or robustness of a system.

Based on the work of Zhu and Ting,²⁷ a novel sensitivity analysis–based DPCE method is proposed in this study. Their objectives are to present a robust parameter design technique based on the theory of performance sensitivity distribution. And the performance index of a system is defined as the ratio of the volume of safe space and real space, which ignores the position relationship between these two spaces. In this work, the focused issue is to evaluate the process capability for small batch production runs. And the capability index of a machining process is measured by considering not only the size of the safe space, but also the position relationship between safe variation space and real variation space. Additionally, a WLS-SVM-based approach to modeling error propagation for machining process is proposed in this study.

Logic framework of DPCE based on sensitivity analysis

Understanding machining error variation from the perspective of sensitivity

It is well known that machining errors are affected by the couple effect of the geometrical errors of machining system and the errors caused by cutting force, heat and vibration. If the machining system is seen as a black box, these errors can be categorized into two classes. One is referred to as system errors, which include geometrical errors of machine tool components and the errors caused by cutting force, heat and vibration. The other can be referred to as input errors, which include the geometrical errors of workpiece (machining error in previous stage), datum errors, fixture errors and installation errors. Usually, input errors have tolerance limits and can be easily measured and controlled in machining processes. System errors are relatively stable during a period of time and cannot be easily inspected and controlled. When there are no assignable causes, machining process is in statistical control state and machining errors are not sensitive to the input errors relatively. When some assignable causes occur, machining process is out of control and machining errors are sensitive to input errors. Consequently, the sensitivity of machining errors against input errors reveals whether the machining process is in control or not to some extent.

Logic framework of sensitivity analysis–based DPCE

Based on the cognition mentioned above, a novel DPCE method based on sensitivity analysis is proposed. As shown in Figure 1, its logic framework mainly contains three steps that are addressed in detail as follows:

Step 1: The error propagation relationships between machining errors and input errors are modeled based on WLS-SVM using the latest history data. The key problems include determining the structure of WLS-SVM model, selection of kernel function, weight assignment and training WLS-SVM.

Step 2: The sensitivity equations of machining errors against input errors are solved according to the regression model constructed in step 1. Then, the error sensitivity distribution against input errors is analyzed and characterized by a set of eigenvalues and eigenvectors in the variation space of input errors.

Step 3: The safe variation space of input errors is depicted as an ellipsoid according to the tolerance limits of quality characteristics. Then, dynamic process capability is evaluated by comparing the fitness between the safe variation space and tolerance space of input errors, and one dynamic PCI is defined. After that, the procedures of calculating the proposed dynamic PCI are stated.

Figure 1.

Logic framework of DPCE based on sensitivity analysis.

In the following three sections, the three key enable technologies, namely construction of WLS-SVM-based error propagation model, sensitivity analysis of machining errors and evaluation of process capability based on sensitivity are addressed, respectively.

Construction of machining error propagation model based on WLS-SVM

Modeling machining error propagation relationships is the foundation of error sensitivity analysis. The performance of error propagation model directly affects the accuracy of sensitivity analysis of machining error as well as process capability evaluation. Least squares support vector machine (LS-SVM) has excellent performance of modeling nonlinear relationship even in the case of small sample size.^34,35 We can obtain the optimal estimate of LS-SVM regression model under the assumption that residual errors of estimate function follow normal distribution.³⁶ However, the assumption is often not realistic in small batch production runs. Therefore, WLS-SVM is employed to construct a more robust regression model describing the complicated error propagation relationships between machining errors and input errors. The principle and mathematical formulation of WLS-SVM can be referred to the work of Suykens et al.³⁷

Determining the structure of WLS-SVM model

The purpose of constructing error propagation model is to analyze the sensitivity of machining errors against input errors, in which the machining system is treated as a black box. The inputs of WLS-SVM model include machining errors in previous stage, datum error, fixture error and install errors. And its output is the machining error of a quality characteristic, such as diameter error, roundness error and cylindricity error. Because a SVM model has only one output, it is necessary to construct a WLS-SVM-based error propagation model for each quality characteristic. The WLS-SVM-based error propagation model of a quality characteristic is shown in Figure 2. The input vector can be formulated as $x = (x_{1}, x_{2}, \dots, x_{n})^{T}$ ; n denotes the number of input errors. The output is denoted as $y_{k} (k = 1, 2, \dots, m)$ ; m represents the number of quality characteristics.

Figure 2.

WLS-SVM-based error propagation model of a quality characteristic.

Selection of kernel function

The performance of SVMs relies on the kernel function and its parameters. Therefore, an appropriate kernel function should be selected while constructing SVM models. There are two most popular kernel functions. One is polynomial function, and the other is Gaussian radial basis function (RBF). Among the two common kernel functions, RBF has more excellent generalization performance in handling nonlinear problems and is the most used kernel function for classification and function estimation.³⁸ Hence, it is selected as the kernel function of WLS-SVM-based error propagation model. According to the mathematical forms of WLS-SVM and RBF, the error propagation model between the kth quality characteristic and its input errors can be formulated as follows

y_{k} = f_{k} (x) = \sum_{i = 1}^{N} α_{k, i} \exp (- \frac{{‖ x_{i} - x ‖}^{2}}{σ_{k}^{2}}) + b_{k}

(1)

where N denotes the number of support vectors, $x_{i}$ denotes the ith support vector. $α_{k, i}$ , $σ_{k}$ and $b_{k}$ , respectively, denote the ith Lagrange multiplier, width parameters of Gaussian kernel function and bias item of the kth quality characteristic’s error propagation model.

Weight assignment

The weights of training samples can also be calculated using weighting functions, such as Huber function³⁹ and Hampel function⁴⁰, according to the model error. Hereinto, Hampel weighting function shows exciting performance in robust estimation, especially in the case that there are extreme outliers in training samples. Consequently, Hampel function is chosen to assign the weights for training data. Thus, the weight of a training sample can be determined according to the residual errors of LS-SVM model which are trained by the same training data

μ_{i} = {\begin{matrix} 1 & | e_{i} / \hat{s} | ⩽ c_{1} \\ \frac{c_{2} - | e_{i} / \hat{s} |}{c_{2} - c_{1}} & c_{1} ⩽ | e_{i} / \hat{s} | ⩽ c_{2} \\ 10^{- 4} & | e_{i} / \hat{s} | ⩾ c_{2} \end{matrix}

(2)

where $μ_{i}$ is the weight of the ith sample, $e_{i}$ denotes the residual error $e_{i}$ corresponding to the ith sample, $c_{1}$ and $c_{2}$ are two constants and $\hat{s}$ is the robust estimate of the standard deviation of residual errors, and its mathematical formula is as follows

\hat{s} = \frac{Q_{d}}{2 \times 0.6745}

(3)

where $Q_{d}$ denotes the quartile deviation and its value is equal to the difference between 75th percentile and 25th percentile of all residual errors. When residual errors follow normal distribution, the probability of $e_{i}$ being greater than 2.5 $\hat{s}$ is very small. Consequently, the values of $c_{1}$ and $c_{2}$ are usually set as 2.5 and 3, respectively.

Training WLS-SVM model

In the practical applications, the historical data of latest finished N workpieces are taken as the training samples after a workpiece is finished. The N samples form an observation window to measure the performance of the current machining process. When the next workpiece is finished, the input error and machining error of another latest finished N workpieces are taken as the training samples by moving the observation window one step toward and so on. A precise error propagation model is the foundation of sensitivity analysis and capability evaluation. And the number of training samples is one of the critical factors influencing the precision of WLS-SVM-based models. However, the error propagation model trained using too many historical data may not be able to reflect the current state of a machining process. To take a compromise between model errors and timeliness, an appropriate number of training samples should be obtained by trial and error. The training procedures are listed as follows:³⁷

Step 1: Train a LS-SVM model through cross-validation using the acquired latest history data ${x_{i}, y_{i}} (i = 1, 2, \dots, N)$ and find an optimal combination $(γ_{k}, σ_{k})$ , where $γ_{k}$ denotes the penalty parameter and $σ_{k}$ denotes the width of RBF, then compute the residual error $e_{i} = α_{k, i} / γ_{k}$ .

Step 2: Calculate $\hat{s}$ according to the distribution of $e_{i}$ .

Step 3: Assign the weight $μ_{i}$ according to equation (2).

Step 4: Solve the quadratic programming equations of WLS-SVM using the same history data ${x_{i}, y_{i}} (i = 1, 2, \dots, N)$ and obtain the WLS-SVM-based error propagation model.

Sensitivity analysis of machining errors

Once the error propagation models are constructed, sensitivity analysis of machining errors can be carried out to evaluate the current stability of the machining process. The procedures of sensitivity analysis on the basis of the models constructed above are detailed in this section.

Sensitivity matrix of machining errors

As mentioned in section “Determining the structure of WLS-SVM model,” suppose the input vector of error propagation model is a n-dimensional vector, namely $x = (x_{1}, x_{2}, \dots, x_{n})^{T}$ , the machining error of the kth quality characteristic is $y_{k}$ . Then, the sensitivity of machining error $y_{k}$ against input error $x_{j}$ can be obtained by computing the partial derivative of $y_{k}$ with respect to $x_{j}$

s_{k, j} = \frac{Δ y_{k}}{Δ x_{j}} = \frac{\partial y_{k}}{\partial x_{j}}

(4)

where $s_{k, j}$ denotes the sensitivity of $y_{k}$ against $x_{j}$ . Substituting equation (1) into (4), $s_{k, j}$ can be formulated as follows

s_{k, j} = \frac{2}{σ_{k}^{2}} \sum_{i = 1}^{N} α_{k, i} \exp (- \frac{{‖ x_{i} - x_{0} ‖}^{2}}{σ_{k}^{2}}) \times (x_{i, j} - x_{0, j})

(5)

where $x_{0, j}$ denotes the nominal value of the jth input error and $x_{0}$ denotes the vector of the nominal values of all input errors. Then, the error sensitivity of all quality characteristics against all input errors can constitute a Jacobian matrix (namely, error sensitivity matrix) with m rows and n columns

J = [\begin{matrix} \frac{\partial y_{1}}{\partial x_{1}} & \frac{\partial y_{1}}{\partial x_{2}} & \dots & \frac{\partial y_{1}}{\partial x_{n}} \\ \frac{\partial y_{2}}{\partial x_{1}} & \frac{\partial y_{2}}{\partial x_{2}} & \dots & \frac{\partial y_{2}}{\partial x_{n}} \\ \dots & \dots & \dots & \dots \\ \frac{\partial y_{m}}{\partial x_{1}} & \frac{\partial y_{m}}{\partial x_{2}} & \dots & \frac{\partial y_{m}}{\partial x_{n}} \end{matrix}]

(6)

Sensitivity distribution of machining errors

The sensitivity of machining errors is analyzed using the theory of performance sensitivity distribution proposed by Zhu and Ting.²⁷ According to the attribute of Jacobian matrix, the machining errors of all quality characteristics can be approximated by the linear model of equation (1)

y = Jx

(7)

where $y = (y_{1}, y_{2}, \dots, y_{m})^{T}$ denotes the machining error vector of m quality characteristics; $J$ denotes the Jacobian matrix calculated at the nominal dimensions.

Obviously, the n input errors are independent. The n-dimensional fluctuation space of these input errors is called as variation space $S_{v}$ .

The relationships between machining errors and input errors can be further depicted by taking a norm of the vector $y$

‖ y ‖^{2} = y_{1}^{2} + y_{2}^{2} + \dots + y_{m}^{2} = y^{T} y = x^{T} J^{T} Jx

(8)

where the operator ‖‖ indicates two-norm. Let $A = J^{T} J$ , then A is a n by n symmetric matrix. Thus, equation (8) can be transformed to

‖ y ‖^{2} = x^{T} Ax

(9)

From the definition of two-norm, it is known that $‖ y ‖^{2}$ is greater than or equal to 0. Thus, the matrix A must be either positive or semi-positive definite. According to the matrix theory, the matrix A has n inter-orthonormal eigenvectors and non-negative eigenvalues $λ_{i} (i = 1, 2, \dots, m)$ . And the number of positive eigenvalues is equal to the rank of A. Sort the eigenvalues in an order from small to large, namely, $0 ⩽ λ_{1} ⩽ λ_{2} ⩽ \dots ⩽ λ_{n}$ , the corresponding eigenvectors is denoted as $v_{i} (i = 1, 2, \dots, m)$ . Matrix B is defined as follows

B = [\begin{matrix} v_{1} & v_{2} & \dots & v_{n} \end{matrix}]

(10)

and diagonal matrix $C = diag (λ_{1}, λ_{2}, \dots, λ_{n})$ . Then, $B$ is an orthogonal matrix and meets $B^{- 1} = B^{T}$ , $A = BC B^{T}$ . Next, the input error vector $x$ is orthogonally transformed through the matrix $B$

x' = B^{T} x

(11)

where $x' = (x'_{1}, x'_{2}, \dots, x'_{n})^{T}$ . According to the property of orthogonal transformation, only the direction and position of vector $x'$ are different from those of $x$ ; their size and other geometry are the same in the variation space. Substituting equation (11) into equation (9) will yield

\begin{array}{l} {‖ y ‖}^{2} = x^{T} B C B^{T} x = {(B^{T} x)}^{T} C (B^{T} x) \\ = {x^{'}}^{T} C x^{'} = λ_{1} {x^{'}}_{1}^{2} + λ_{2} {x^{'}}_{2}^{2} + \dots + λ_{n} {x^{'}}_{n}^{2} \end{array}

(12)

Equation (12) describes the sensitivity distribution of machining error in n-dimensional variation space of input errors. According to the feature of matrix A, it can be divided into two kinds of cases to be, respectively, investigated in detail:

1. If A is a positive definite matrix, then $rank (A) = n, λ_{i} > 0, i = 1, 2, \dots, n$ . Equation (12) can be transformed to

\frac{x'_{1}^{2}}{{(| | y | | / \sqrt{λ_{1}})}^{2}} + \frac{x'_{2}^{2}}{{(| | y | | / \sqrt{λ_{2}})}^{2}} + \dots + \frac{x'_{n}^{2}}{{(| | y | | / \sqrt{λ_{n}})}^{2}} = 1

(13)

Equation (13) defines a family of n-dimensional hyper-ellipsoids. The length of a semi-principal axis $x'_{i}$ is equal to $| | y | | / \sqrt{λ_{i}}$ and its direction is the same as that of the corresponding eigenvector $v_{i}$ .

For a certain value of $| | y | |$ , the corresponding hyper-ellipsoid surface encloses a sub-variation space. Every point $x = (x_{1}, x_{2}, \dots, x_{n})^{T}$ on the ellipsoid surface indicates one vector of the input errors. And different points on the ellipsoid surface result in the same value of $| | y | |$ . Consequently, the larger the value of $| | x | |$ is, the less sensitive the machining error to the input errors is at the direction of the vector $x$ . Obviously, the largest and smallest values of $| | x | |$ correspond to the longest and shortest principal axes. Due to $0 ⩽ λ_{1} ⩽ λ_{2} ⩽ \dots ⩽ λ_{n}$ , machining error is the least sensitive to the input error variation along the direction of $v_{1}$ and the most sensitive to the input error along the direction of $v_{n}$ . An example with three-dimensional input errors is shown in Figure 3(a).

2. If A is a semi-positive definite matrix, then $rank (A) = r < n$ , $λ_{1} = λ_{2} = \dots = λ_{n - r} = 0$ and $0 < λ_{n - r + 1} ⩽ λ_{n - r + 2} ⩽ \dots ⩽ λ_{n}$ . Similarly, equation (12) can be transformed to

\begin{array}{l} \frac{{x^{'}}_{n - r + 1^{2}}}{{(| | y | | / \sqrt{λ_{n - r + 1}})}^{2}} + \frac{{x^{'}}_{n - r + 2^{2}}}{{(| | y | | / \sqrt{λ_{n - r + 2}})}^{2}} \\ + \dots + \frac{{x^{'}}_{n}^{2}}{{(| | y | | / \sqrt{λ_{n}})}^{2}} = 1 \end{array}

(14)

Figure 3.

Sensitivity distribution of machining errors (n = 3): (a) r = 3 and (b) r = 2.

According to the geometric meaning of equation (13), equation (14) can be interpreted as an n-dimensional hyper-cylindroid with $n - r$ infinite principal axes, which can be treated as a particular case of an n-dimensional ellipsoid. As mentioned above, the longest and smallest principal axes correspond to the least sensitive and most sensitive direction of machining error to input errors, respectively. Therefore, the $n - r$ infinite principal axes mean that machining errors are not sensitive at their direction, namely the machining error caused by the factor variations along the directions of theses axes is equal to zero. This seems impractical and even ridiculous. However, the input errors at the insensitive directions are generally relatively small in real machining systems; meanwhile, the linear approximation model described in equation (7) is established under the assumption of small errors as well. Hence, it is not necessary to intuitively doubt the validity of the analysis procedure. Figure 3(b) shows an example of three-dimensional cylindroid with $r = 2$ . The cylindroid has the first principal axis at the $v_{1}$ direction and other two principal axes at the $v_{2}$ and $v_{3}$ directions. According to the analysis results, $v_{1}$ and $v_{3}$ are the least and most sensitive directions of machining errors.

Evaluation of process capability based on sensitivity distribution

Determination of safe variation space

To ensure machining errors be in their corresponding tolerance limits, the machining errors of all quality characteristics need to meet the following constraints

‖ y ‖^{2} ⩽ \sum_{k = 1}^{m} t_{k}^{2} = Y_{t}^{2}

(15)

where $t_{k}$ denotes the tolerance of the kth quality characteristic. And there is a corresponding subspace of $S_{v}$ , in which the resulted machining errors satisfy equation (15). This subspace is called as safe variation space of input errors, which is denoted as $S_{s}$

S_{s} = {x | x^{T} Ax ⩽ Y_{t}^{2}}

(16)

For a machining system, suppose $T = (t_{1, l}, t_{1, u}, t_{2, l}, t_{2, u}, \dots, t_{n, l}, t_{n, u})$ represents the tolerance vector of input errors, $t_{i, l}$ denotes the lower specification limit of the ith input error and $t_{i, u}$ denotes its upper specification limit. At the same time, let $S_{r} {t_{i, l} ⩽ x_{i} ⩽ t_{i, u}, i = 1, 2, \dots, n}$ denotes the true variation space of input errors. Because the input errors are independent each other in the practical production, $S_{r}$ is an n-dimensional block space enclosed by $2 n$ planes. And the length along the ith direction is equal to $t_{i, l} + t_{i, u}$ .

Then, the problem of process capability evaluation can be transformed to comparing the fitness between safe variation and actual variation space, which is defined by the tolerance limits of all input errors. However, only the linearity relationship between machining error vector $y$ and input error vector $x$ described in equation (7) is held; we can ensure the reasonability of the analysis. Thus, it is essential to constrain the magnitude of $| | x | |$ in the safe variation space within a certain range to meet the linearity effectiveness before evaluating the capability of a machining process. This can be carried out by adjusting the lengths of the principal axes that violate the linear relationships.

In general, the linear ranges of different machining systems are not the same. They usually rely on the characteristics of a machining system. Some investigations demonstrated that the linear relationships could be held when the component variations are less than 3%–5% of the nominal dimensions.⁴¹ It should be pointed out that a cylindroid safe variation space will be converted to an ellipsoid safe space after their infinite principal axes are reduced to finite lengths. The principal axis directions of the modified safe space are the same as those of the original safe space, because the error sensitivity distribution reflected by these axes remains the same.

To limit the input error vector $x$ in the corrected safe space $S_{s}$ , a correction coefficient $ξ$ is introduced to rectify the safe variation range of each input error under the current state of a machining system

| x_{i} | ⩽ ξ d_{i} i = 1, 2, \dots, n

(17)

where $d_{i}$ denotes the nominal value of the ith input error. As mentioned above, the value of $ξ$ can be set within the range of 0.03–0.05. Then, every eigenvalue $λ_{i}$ is corrected through

λ'_{i} = \max (λ_{i}, \frac{{Y_{t}}^{2}}{ξ^{2} d_{i}^{2}})

(18)

to amend the corresponding semi-principal axis length in the modified safe space

a_{i} = Y_{t} / \sqrt{{λ'}_{i}} (i = 1, 2, \dots, n)

(19)

where $a_{i}$ denotes the length of the ith semi-principal axis. And the principal axis directions keep unchanged. In this case, every semi-principal axis length is not larger than $ξ d_{i}$ ; therefore, the modified safe space can be regarded as a linear error propagation space.

The following discussions mainly aim at the corrected safe variation space rather than the original one. For avoiding confusion, $S_{s}$ denotes the corrected safe variation space that will be simply called as the safe space, unless there are special declarations.

Definition of dynamic PCI

According to the definition of process capability, we can measure the performance of a process by comparing the fitness between the safe variation space and the tolerance space of input errors. However, directly using the ratio of the volume of safe variation space to that of real variation space as a measure of process capability may result in error prone conclusions, because the direction of each principal axis also affects the amount of the safe variation space intersecting with the real space. Consequently, the capability of a machining process should be measured by considering not only the size of the safe space but also the directions of the principal axes, namely taking the position relationship between safe variation space and real variation space into considerations to reflect the actual process capability.

The safe variation space is a hyper-ellipsoid and the directions of their principal axis may not coincide with those of the coordinate axis, while the real variation space is a block. Therefore, the position relationship between them may include the following four situations, as shown in Figure 4 ( $n = 2$ ). (1) The real variation space completely falls inside the safe space, namely $S_{r} \subseteq S_{f}$ , as shown in Figure 4(a). In this case, the process capability is adequate. (2) The real variation space completely encircles the safe space, namely $S_{f} \subseteq S_{r}$ , as shown in Figure 4(b). In this case, the process capability is not adequate. (3) There is no inclusion relation between the real variation space and the safe space and the intersection between them is not null, namely, $S_{r} ⊄ S_{f}$ , $S_{f} ⊄ S_{r}$ and $S_{f} \cap S_{r} \neq \emptyset$ . This reveals that the process capability is also not adequate. (4) The intersection between the real variation space and the safe space is null, namely, $S_{f} \cap S_{r} = \emptyset$ . This situation means that the machining process is extremely unstable and the process output falls out the specification limits certainly. Yet, this situation usually does not occur in the practical production.

Figure 4.

Position relationship between safe space and real variation space (n = 2): (a) S_r ⊆ S_s, (b) S_s ⊆ S_r and (c) S_r ⊆ S_s, S_s ⊆ S_r and S_r ∩ S_s ≠ ∅.

From the above, a new PCI $C_{ps}$ is put forward and defined as the ratio of the volume of the intersection between these two spaces to that of the real space, as formulated in equation (20), under the assumption that these input errors are independent and change randomly in predetermined tolerance specifications

C_{ps} = {\begin{matrix} \frac{V_{{S'}_{r}}}{V_{S_{r}}} S_{r} \subseteq S_{s} \\ \frac{V_{S_{I}}}{V_{S_{r}}} S_{r} ⊄ S_{s} \end{matrix}

(20)

where $S'_{r}$ denotes the equidistant offset block space from $S_{r}$ , which is the least block inscribed with the safe space $S_{s}$ . Its geometric representation is shown in Figure 5 in a case of two-dimensional variation space of original errors. $S_{I}$ denotes the intersection between $S_{r}$ and $S_{s}$ ; $V_{{S'}_{r}}$ , $V_{S_{r}}$ and $V_{S_{I}}$ denote their volumes, respectively.

Figure 5.

Equidistant offset block space $S'_{r}$ from $S_{r}$ .

If the outputs of a machining process are not relatively sensitive to input error variation, the process capability is adequate. The more adequate is the process capability, the larger is the size of $S'_{r}$ and the value of $C_{ps}$ .

The traditional PCIs are defined under the assumption that the machining process is in statistical control. The proposed PCI is defined for the processes which are in dynamically stable state and reflects the capability of the processes ensuring machining errors in related tolerance limits. Therefore, it is called as dynamic PCI.

Calculation of the proposed PCI

To compute the value of $C_{ps}$ , the volumes of $S'_{r}$ , $S_{r}$ and $S_{I}$ must be calculated first. However, it is very hard to directly judge the position relationship between the safe space and the real space through geometrical analysis. Especially when the dimension of variation space is larger than 3, it is impossible to attain the analytical solutions for the volumes of the intersection $S'_{r}$ and $S_{I}$ . Then, we can only get its approximate solution through numerical method. The detailed calculation procedures include seven steps:

Step 1: Along the direction of each coordinate axis, discretize the real variation space into finite n-dimensional block units with a certain step h and compute the center coordinate of each unit.

Step 2: Judge whether each block unit falls inside the safe variation space one by one according to equation (21) and cumulate the volumes of all space units that fall inside the safe variation space to gain the volume of the intersection $S_{I}$ , where the volume of a block unit is equal to $h^{n}$

\frac{x'_{1}^{2}}{a_{1}^{2}} + \frac{x'_{2}^{2}}{a_{2}^{2}} + \dots + \frac{x'_{i}^{2}}{a_{i}^{2}} + \dots + \frac{x'_{n}^{2}}{a_{n}^{2}} ⩽ 1

(21)

where $a_{i}$ denotes the length of the ith semi-principal axis, as defined in equation (19).

Step 3: Compare the values between $V_{S_{I}}$ and $V_{S_{r}}$ which is calculated according to equation (22). If the value of $V_{S_{I}}$ is larger than that of $V_{S_{r}}$ , it will turn to step 6. Conversely, it will turn to step 4

V_{S_{r}} = Π_{i = 1}^{n} (t_{i, u} - t_{i, l})

(22)

Step 4: Initialize the value of $V_{{S'}_{r}}$ as that of $V_{S_{r}}$ and construct a temporary block space $S_{r}^{t} {t_{i, l}^{t} ⩽ x_{i} ⩽ t_{i, u}^{t}, i = 1, 2, \dots, n}$ which is equal to $S_{r}$ .

Step 5: Construct a equidistant offset block space $S'_{r} {t'_{i, l} ⩽ x_{i} ⩽ t'_{i, u}, i = 1, 2, \dots, n}$ from the block space $S_{r}^{t} {t_{i, l}^{t} ⩽ x_{i} ⩽ t_{i, u}^{t}, i = 1, 2, \dots, n}$ with a distance h, namely, $t'_{i, l} = t_{i, l}^{t} - h$ , $t'_{i, u} = t_{i, u}^{t} + h$ , and compute the vertex coordinates of $S'_{r}$ .

Step 6: Judge whether each vertex of $S'_{r}$ falls inside $S_{s}$ one by one according to equation (21). If all of the vertexes lie within $S_{s}$ , compute the volume of $S'_{r}$ similarly with $S_{r}$ and make $S_{r}^{t}$ be equal to $S'_{r}$ , then turn to step 4. Conversely, it will turn to step 7.

Step 7: Compute the value of the index $C_{ps}$ according to equation (20).

The general idea behind a PCI is to compare what a process “should do” with what it “is actually doing.”⁴² In the classic definitions of PCIs, the tolerance band of a quality characteristic is what a process “should do” and the process variability and centering are what it “is actually doing.” Whereas in the proposed PCI, the real variation space indicates what a process “should do” and the safe space indicates what it “is actually doing.” It should be pointed out that the physical significance of the PCI is similar with that of the classic PCIs $C_{p}$ and $C_{pk}$ but not the same entirely. For example, the classic index $C_{pk} = 1.33$ implies the probability of non-conforming items is equal to 0.01% for the normal distribution. Usually, a process is considered to be capable when the value of the classic $C_{pk}$ is larger than or equal to 1.33. While for the proposed index $C_{ps}$ , a process can be believed to have plenty of capability as long as its value is larger than or equal to 1, which implies that the real variation space falls inside the safe space completely.

Case study and discussions

In this section, a practical case is addressed to demonstrate the feasibility and effectiveness of the proposed approach for process capability evaluation. The corresponding computing program is coded in MATLAB and runs on a DELL OPTIPLEX 360 with a 2.93 GHz processor and 2 GB random access memory (RAM). Finally, some discussions are also stated.

A practical case

The deep-hole boring process for the outer barrel part of an airplane landing gear is taken as an example. And the drawing of the outer barrel part is shown in Figure 6. The key quality characteristics include diameter, roundness and coaxiality, which are dimensioned with red color, and their specification limits are [0, 0.04], [0, 0.02] and [0, 0.03], respectively. The positioning datum are outer cylinder A, outer cylinder B and left end face. Consequently, the corresponding input errors of machining process include the alignment error of datum A and datum B, the diameter error and roundness error in the previous stage. Their allowed tolerance limits are [0, 0.05], [0, 0.05], [−0.1, 0.1] and [0, 0.04], respectively. Because of the customized trait of airplane products, the lot size of outer barrel parts in a batch production run is commonly small and ranges usually from ten to several tens. Therefore, their machining processes are typical small batch production runs. Meanwhile, the cost of each blank is very high, which ranges from 20,000 to tens of thousands dollars. A quality accident may result in huge financial losses. Consequently, it is necessary to real-time implement process capability evaluation while machining this type of parts. In a small batch production, the input errors and corresponding machining errors of sequential 25 workpieces are acquired, as shown in Table 1. To observe the fluctuation of the process capability, a window size 20 is adopted to construct the error propagation models for three key quality characteristics and evaluate the current process performance. The analysis procedures for the process capability are stated as follows:

Step 1: Establish the error propagation model based on WLS-SVM. To find a compromise size to balance model errors and timeliness, one experiment has been conducted to compare different sizes of observation window. Different window sizes with 12, 16, 20 and 24 samples are considered. The mean absolute relative error (MARE) is taken as evaluation criterion. The MATLAB LS-SVM Toolbox by De Brabanter et al.⁴³ was used to construct WLS-SVM-based error propagation model and evaluate the performance of different window sizes. The comparison results are presented in Table 2. According to the results, the window size with 20 and 24 samples produced almost the same MARE (4.78% and 4.75%), and there is no significant improvement on MARE from the window size with 20–24. Therefore, the window size 20 is selected in this application case.

Figure 6.

Drawing of the outer barrel part.

Table 1.

Acquired input errors and machining errors of sequential 25 workpieces.

Part no.	Alignment error at datum A	Alignment error at datum B	Diameter error in previous stage	Roundness error in previous stage	Diameter error	Roundness error	Coaxiality error
1	0.019	0.032	0.05	0.021	0.015	0.015	0.017
2	0.030	0.034	−0.08	0.025	0.007	0.014	0.019
3	0.015	0.022	0.02	0.016	0.021	0.012	0.014
4	0.031	0.015	0.01	0.011	0.026	0.010	0.020
5	0.018	0.015	−0.06	0.023	0.008	0.015	0.028
6	0.031	0.023	0.03	0.022	0.024	0.013	0.023
7	0.038	0.020	−0.03	0.017	0.026	0.012	0.023
8	0.006	0.019	0.01	0.013	0.022	0.011	0.009
9	0.011	0.018	−0.07	0.016	0.001	0.011	0.012
10	0.026	0.024	−0.06	0.027	0.012	0.015	0.027
11	0.034	0.036	−0.01	0.017	0.033	0.013	0.015
12	0.021	0.020	0.03	0.015	0.019	0.012	0.017
13	0.013	0.023	0.02	0.016	0.021	0.012	0.012
14	0.023	0.038	−0.02	0.019	0.029	0.013	0.010
15	0.019	0.021	0.04	0.022	0.017	0.013	0.020
16	0.024	0.021	0.09	0.016	0.002	0.012	0.016
17	0.029	0.017	−0.01	0.020	0.028	0.012	0.022
18	0.019	0.025	0.02	0.009	0.021	0.011	0.011
19	0.031	0.009	0.05	0.019	0.012	0.011	0.024
20	0.026	0.023	0.02	0.017	0.026	0.012	0.018
21	0.025	0.032	−0.04	0.014	0.021	0.009	0.016
22	0.038	0.056	0.09	0.013	0.013	0.012	0.011
23	0.036	0.025	0.01	0.017	0.007	0.011	0.013
24	0.019	0.027	−0.03	0.013	0.016	0.008	0.012
25	0.023	0.012	0.06	0.033	0.024	0.012	0.015

Table 2.

Performance comparison for different window sizes.

Window size	12	16	20	24
MARE (%)	12.62	9.58	4.78	4.75

Afterward, the observations of the first 20 workpieces are taken as the training samples of WLS-SVM-based error propagation models for three key quality characteristics. Then, another three error models are constructed by moving the observation window one step toward and so on. Moving the observation window from part no. 20 to 24 successively, 15 regression models for three key quality characteristics are constructed, respectively. The hyper-parameters $(γ, σ)$ are determined by means of fourfold cross-validation on the training data. The obtained hyper-parameters of the 15 models are reported in Table 3.

Step 2: Solve the Jacobian matrix. According to equations (5) and (6), the Jacobian matrix $J$ composed of the sensitivities of three key quality characteristics to four input errors is solved from the error propagation models established in step 1, which are reported in Table 4.

Step 3: Construct the error propagation characteristic matrix and calculate the eigenvalues and eigenvectors. The error propagation characteristic matrix can be attained according to the equation $A = J^{T} J$ . Then, their four eigenvalues $λ_{i}$ and four eigenvectors $v_{i}$ can be calculated. These eigenvalues are reported in Table 5.

Step 4: Modify the safe variation space and calculate the eigenvalues of the safe space. According to equation (15), the value of $Y_{t}^{2}$ can be gained

Y_{t}^{2} = \sum_{k = 1}^{3} t_{k}^{2} = 0 . 04^{2} + 0 . 02^{2} + 0 . 03^{2} = 0.0029

Table 3.

Obtained hyper-parameters of 15 WLS-SVM-based error propagation models.

Window no.	Quality characteristics	$γ$	$σ$
1	Diameter	4346.8373	5.7553
	Roundness	3606.6703	14.0355
	Coaxiality	7322.1340	5.4763
2	Diameter	92.7749	59.4011
	Roundness	104.6589	5.1042
	Coaxiality	4205.7111	8.7760
3	Diameter	548.9422	7.1464
	Roundness	20.1327	3.4670
	Coaxiality	496.1846	3.8125
4	Diameter	587.1648	7.2169
	Roundness	25.4155	8.9897
	Coaxiality	455.4810	4.8391
5	Diameter	2057.6507	5.2539
	Roundness	5.5070	3.2628
	Coaxiality	2962.4003	10.4863

Table 4.

Sensitivity of quality characteristics to input errors.

Window no.	Quality characteristics	Alignment error at datum A	Alignment error at datum B	Diameter error in previous stage	Roundness error in previous stage
1	Diameter error	0.0045	0.2124	−0.2739	0.2176
	Roundness	0.0989	0.0871	0.0467	0.2615
	Coaxiality	0.2178	0.1559	−0.0626	0.0658
2	Diameter error	0.0025	0.2287	−0.2514	0.2062
	Roundness	0.1344	0.0772	0.0258	0.2909
	Coaxiality	0.2539	0.1697	−0.0684	0.0772
3	Diameter error	0.0032	0.2337	−0.2265	0.1979
	Roundness	0.1176	0.0584	0.071	0.2792
	Coaxiality	0.2396	0.1967	−0.1723	0.0705
4	Diameter error	0.0027	0.2404	−0.2652	0.2082
	Roundness	0.1241	0.0677	−0.1071	0.2654
	Coaxiality	0.2522	0.1775	−0.0723	0.0627
5	Diameter error	0.0340	0.2337	−0.2772	0.1979
	Roundness	0.1176	0.0584	−0.0382	0.2792
	Coaxiality	0.1996	0.1967	−0.0705	0.0904

Table 5.

Eigenvalues of error propagation characteristic matrix A.

Window no.	$λ_{2}$	$λ_{3}$	$λ_{4}$
1	0.0338	0.0586	0.2430
2	0.0385	0.0643	0.2685
3	0.0379	0.0702	0.2680
4	0.0264	0.0553	0.2960
5	0.0292	0.0435	0.2875

From Figure 6, it is known that both the nominal values of the first two input errors are equal to 175. And it is known that both the nominal values of the last two input errors are equal to 144.8 according to its technique documents. Let the modification coefficient $ξ = 0.03$ . Then, the corresponding eigenvalues $λ'_{i} (i = 1, 2, 3, 4)$ can be gained according to equation (18), and they are list in Table 6. Then the modified safe variation space is obtained and the corresponding lengths of semi-principal axes are reported in Table 7. It can be seen that all of the semi-principal axis lengths at the most sensitive direction, namely $a_{4} s$ , are much larger than the allowed tolerance limit of the fourth input error, which shows the most sensitive direction has adequate safe variation range.

Step 5: Calculate the volumes of $S_{r}$ , $S_{I}$ and $S'_{r}$ , and the capability index $C_{ps}$ . From equation (22), the volume of the real variation space can be calculated as follows

\begin{array}{l} V_{S_{r}} = \prod_{i = 1}^{n} (t_{i, u} - t_{i, l}) = 0.05 \times 0.05 \\ \times [0.1 - (- 0.1)] \times 0.04 = 0.00002 \end{array}

Table 6.

Modified eigenvalues of characteristic matrix A with $ζ = 0.03$ .

Window no.	$λ'_{1}$	$λ'_{2}$	$λ'_{3}$	$λ'_{4}$
1	1.0522E−04	0.0338	0.0586	0.2430
2	1.0522E−04	0.0385	0.0643	0.2685
3	1.0522E−04	0.0379	0.0702	0.2680
4	1.0522E−04	0.0264	0.0553	0.2960
5	1.0522E−04	0.0292	0.0435	0.2875

Table 7.

Lengths of the semi-principal axes.

Window no.	$a_{1}$	$a_{2}$	$a_{3}$	$a_{4}$
1	5.2500	0.2928	0.2224	0.1092
2	5.2500	0.2746	0.2124	0.1039
3	5.2500	0.2767	0.2032	0.1040
4	5.2500	0.3315	0.2291	0.0990
5	5.2500	0.3153	0.2583	0.1004

According to the calculation procedures of the proposed PCI, we can calculate the volumes of the intersection between the safe space and real variation space, and the volume of the equidistant offset block from the real space with a step 0.001. Then, the capability index $C_{ps}$ can be attained from equation (20). These calculated values within five observation windows are listed in Table 8.

Table 8.

Values of $C_{ps}$ corresponding to different windows.

Window no.	$V_{S_{I}}$	$V_{{S'}_{r}}$	$C_{ps}$
1	0.00002	3.3586E−05	1.3793
2	0.00002	3.5780E−05	1.4890
3	0.00002	2.6174E−05	1.3087
4	0.00002	2.2776E−05	1.2388
5	0.00002	2.2852E−05	1.2426

The run chart of the proposed PCI is plotted in Figure 7, from which two phenomena can be seen: (1) all of the values of $C_{ps}$ are larger than 1, which indicates the current process has plenty of capability to bore the deep-hole of the outer barrel part; (2) the value of $C_{ps}$ fluctuates within certain ranges with the production run, which means that the machining process is not stable but capable, namely dynamically stable. From Table 1, it is found that all of the machining errors of the workpieces no. 21–25 fall in tolerance limits. This shows the measure of the boring process using the proposed method is reasonable.

Figure 7.

Run chart of the proposed PCI.

Discussion

The five steps mentioned above show the feasibility of using the proposed method to evaluate the capability of a real small batch production run in a job-shop. It also indicates that the sensitivity-based process capability evaluation method is systematic and effective, because the five steps transform one by one smoothly and logically. Among these steps, the error propagation model is constructed using statistical learning method (WLS-SVM) without normality detection and distribution parameters estimation, which need to be done in traditional process capability evaluation methodologies; the latter four steps are carried out on the foundation of the first step and also need not the assumption conditions. It clearly reveals how to evaluate the performance of a small batch production run and avoid the required assumptions for SPC techniques. Unlike the traditional PCIs, the proposed PCI measures the process capability from a new perspective of the sensitivity of machining errors. Although the physical meanings of the proposed process performance indices are not similar with those of traditional PCIs, they also reveal the process performance quantitatively by relating its capability to the predetermined specification limits. Consequently, the proposed method can be used to perform process capability analysis and get rid of the influence caused by the assumptions which are the foundations of traditional capability evaluation methods but are commonly violated in real machining processes.

It should be pointed out that constructing error propagation model plays a critical role in the presented method and the model precision directly affects the evaluating result of process capability. However, the number of training data influences the performance of a WLS-SVM regression model significantly. Insufficient training data may generate imprecise error propagation model and lead to error conclusions of process performance. When the number of workpieces in a production batch is very small or the machining process is at starting stage, the available observation data are rare limited and usually less than 20. In this case, we can adopt the first-order terms of Taylor series expansion²⁸ to establish the linear approximation model of error propagation relationships, so as to evaluate the process capability based on sensitivity analysis.

Although the above practical case focuses mainly on a small batch production run, the sensitivity analysis–based process capability evaluation method can also be applied in mass production without any change, in which we can acquire plenty of observation data to train WLS-SVM-based error propagation. Moreover, the proposed method for evaluating process capability in a single machining process is the base of further measuring the performance of multi-stage machining processes.

In all, this work attempts to evaluate the process performance for small batch production runs from an entirely different viewpoint to terminate the trouble resulted by the assumption of independent and identical distribution. And the results of application case show that the method is promising.

Conclusion

A sensitivity analysis–based process capability evaluation method for small batch production runs is proposed in this article, and a corresponding capability index is also presented. In this method, an error propagation model between machining errors and input errors is first established using WLS-SVM. Then, the sensitivity distribution of machining errors versus input errors is characterized by a set of eigenvalues and eigenvectors in the variation space of input errors. Finally, the safe variation space of input errors is depicted as an ellipsoid and the process capability is evaluated by comparing the fitness between the safe variation space and the tolerance space of input errors. The results of a practical case demonstrate that the proposed method is feasible and effective in real world and does not need the assumption of independent and identical distribution, which are essential for traditional process capability evaluation methods. In a word, this work is an attempt to measure the performance of small batch production runs from a novel perspective of the sensitivity of machining errors, and the results of an application case demonstrate that it is promising.

In the future, one important work is to investigate the relationship between the proposed PCI and process yield, which will further promote its practicability. Another ongoing work is to further investigate how to evaluate the performance of multi-stage machining processes based on the proposed method.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research work is supported by the National Basic Research Program of China (973 program) with grant No. 2011CB706805. The authors hereby thank the MOST of China for the financial aids.

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Sensitivity analysis–based dynamic process capability evaluation for small batch production runs

Abstract

Keywords

Introduction

Literature review

Logic framework of DPCE based on sensitivity analysis

Understanding machining error variation from the perspective of sensitivity

Logic framework of sensitivity analysis–based DPCE

Construction of machining error propagation model based on WLS-SVM

Determining the structure of WLS-SVM model

Selection of kernel function

Weight assignment

Training WLS-SVM model

Sensitivity analysis of machining errors

Sensitivity matrix of machining errors

Sensitivity distribution of machining errors

Evaluation of process capability based on sensitivity distribution

Determination of safe variation space

Definition of dynamic PCI

Calculation of the proposed PCI

Case study and discussions

A practical case

Discussion

Conclusion

Footnotes

Declaration of Conflicting Interests

Funding

References