Fuzzy multivariate process capability vector

Abstract

Process capability indices are numerical measures used to determine whether the process products meet the quality requirement dedicated by the customers and producers or not. When the products’ quality counts on two or more correlated characteristics, multivariate process capability indices are applied. In this paper, we focus on two indices proposed by Shahriari and Abdollahzadeh (2009) and Taam et al. (1993). The traditional multivariate capability indices can’t be applied if the specification limits and target value of each characteristic are presented in a linguistic form. Instead, the fuzzy set theory is used to deal with these circumstances. Here, the fuzzy formulations of the indices are developed. Furthermore, numerical examples are presented to illustrate the applicability of the proposed methods and we make a comparison between them.

Keywords

Multivariate normal distribuition multivariate capability index fuzzy logic triangular fuzzy number trapezoidal fuzzy quantity

1 Introduction

A process is a unique combination of tools, materials, methods, and people employed in producing a measurable yield; for example a manufacturing line of machine parts. All processes have an inherent statistical variability which can be studied by statistical methods. The output of a process is expected to meet the customers’ demands, specifications, or engineering tolerances. Engineers can conduct a process capability investigation to find out the extent to which the process can meet these expectations.

The power of a process to introduce high quality products can be evaluated by control charts and capability indices. Using control charts involves running the process to obtain enough measurable outputs to ensure the engineering that the process is stable and so its mean and variability can be reliably estimated. Statistical process control defines techniques to properly differentiate between stable processes, processes that are drifting (experiencing a long-term change in the mean of the output), and processes that are growing more variable. It is noted that process capability indices (PCIs) are only meaningful for processes that are stable (in a state of statistical control).

The aim of capability analysis is to ensure that a process is capable to meet the customers’ specifications, and we use capability indices to make that assessment. Process capability is a numerical measure to find out the measurable property of a process to a specification. In the cases that the products possess one characteristic, univariate process capability indices are used, but when the products’ quality depends on two or more characteristics, it is essential to apply the multivariate process capability indices.

Recently, there have been more researches dedicated to multivariate process capability indices such as Chan et al. [3], Hubele et al. [9], Taam et al. [36], Chen [4], Shahriari and Lawrence [34], Wang and Chen [37], Wang and Du [38], Chen et al. [5], Castagliola et al. [2], Shahriari and Abdollahzadeh [33], Goethals and Cho [7], Pan and Wendy [22], Pan and Chung-I Li [23], Zhang Min et al. [41].

In many industries, manufactured products are classified into two groups: conforming and non-conforming. A product is said to be conforming if its quality characteristic measurement falls within the specification limits. Otherwise, it is said to be non-conforming. It is generally assumed that a given unit of product either does or does not conform to the specifications. Traditional tolerance interval have contributed to this notion.

A graded specification limit is an index that may be employed in the sense of quality to deal with the products in which their quality characteristic could be reworked and refined when they are non-conforming to a certain extent. This way is economical, especially when the reworking cost is much less than the wasting cost. A graded representation of the specification limits is handled by employing the theory of fuzzy sets. In addition, fuzzy set theory is used to add more information and flexibility to PCIs than the traditional ones. Charles W. Bradshaw, Jr. [1] presented an excellent explanation in the field of fuzzy process control, particularly on the specifications’ limits. He has given some reasons for considering specification limits as fuzzy numbers, (for more information see [1]). The reason for using fuzzy target value is as the same as the above reason. In the literature, there have been many articles in the field of quality control in which the specification limits and target value are assumed to be fuzzy. For example, Kaya and Kahraman [17] assumed the situation in which the specification limits and target value cannot be defined by using crisp numbers, so, they applied fuzzy set theory. Furthermore, Moeti et al. [20], Parchami and Mashinchi [25, 26] and Parchami et al. [24] consider specification limits and target value imprecise numbers so they are expressed in fuzzy terms.

In the literature one can find numerous papers which aggregate univariate capability indices with the fuzzy set theory. Papers such as Yongting [40], Lee et al. [18], Lee [19], Sadeghpour-Gildeh [29, 30], Hsu and Shu [8], Parchami and Mashinchi [27], Parchami et al. [28], Kahraman and Kaya [11, 12] and Kaya and Kahraman [13 –16]. On the contrary multivariate capability indices have received much less attention in this field.

In multivariate case, Sadeghpour-Gildeh and Moradi [31] considered fuzzy tolerance region and introduced fuzzy multivariate capability index, directly assessed the proportion of future parts which may be expected to lie outside the tolerance region and it does not pay attention to the process centering.

In this paper, we focus on the multivariate process capability indices introduced by Taam et al. [36] and by Shahriari and Abdollahzadeh [33]. We consider a situation in which the specification limits and the target for each characteristic are not precise. So, these objects are described as fuzzy numbers (quantities). Since a crisp multivariate capability is not adequate, it will be fuzzy. Here is our paper structure. In the subsequent section, we refer to the multivariate process capability indices proposed by Taam et al. [36] and by Shahriari and Abdollahzadeh [33]. In Section 3, we discuss fuzzy logic and review some basic definitions. Fuzzy multivariate capability indices are introduced in Section 4. We apply our proposed indices to obtain the capability of two processes in two examples in Section 5. Finally, Section 6 represents the conclusions.

2 Multivariate process capability indices

Let’s consider the quality of the products counts on p correlated characteristics. These characteristics are multivariate normally distributed by mean vector μ and variance-covariance matrix Σ. Suppose μ and Σ are unknown. So, they should be estimated from the data gathered of the in-control process. Let X₁, X₂, . . . , X_n indicate a random sample vector of size n from this process. The sample mean vector, $\bar{X}$ contains the sample means of the n observations for the p characteristics and the sample variance-covariance matrix, S, a p × p matrix contains the sample variances and covariances of the characteristics. Therefore $\bar{X}$ estimates μ and S estimates Σ.

Taam et al. [36] defined a multivariate capability index as a ratio of two volumes ${MC}_{pm} = \frac{vol . (R_{1})}{vol . (R_{2})},$ (1)where R₁ is a modified tolerance region and R₂ is a scaled 99.73% process region (scaled by the process mean square error), that is, an elliptical region represented by the quadratic form $(X - μ)^{'} Σ_{r}^{- 1} (X - μ) \leq χ_{p, 0.0027}^{2}$ , where $χ_{p, 0.0027}^{2}$ is the upper 0.0027 percentile of chi-square distribution with p degrees of freedom. So, MC_pm can be written as ${MC}_{pm} = \frac{vol . (R_{1})}{vol . (R_{3})} \times \frac{1}{D} = ψ_{p} \times D^{- 1},$ (2)where R₃ is the region in which 99.73% of the process values fall within and D = [1 + ( μ - T) ′Σ^-1 ( μ - T)] ^1/2. ψ_p represents the process variability relative to the modified tolerance region and D reflects the process deviation from target T.

Based on the gathered samples, the estimator for the index MC_pm is given as ${\hat{MC}}_{pm} = \frac{{\hat{MC}}_{p}}{\hat{D}},$ (3)where ${\hat{MC}}_{p} = \frac{vol . (modified tolerance region)}{∣ S ∣^{1 / 2} (π χ_{p, 0.0027}^{2})^{p / 2} [Γ (p / 2 + 1)]^{- 1}},$ (4)and $\hat{D} = [1 + \frac{n}{n - 1} (\bar{X} - T)^{'} S^{- 1} (\bar{X} - T)]^{1 / 2} .$ (5)

Modified tolerance region is the largest ellipsoid centered at the target completely within the original tolerance region. The process will be capable if ${\hat{MC}}_{pm}$ is greater than 1.

Shahriari and Abdollahzadeh [33] introduced the multivariate capability vector consisting three components as the following $NMPCV = [{NMC}_{pm}, PV, LI],$ (6)where the first component is the ratio of the “modified tolerance region” on the region that covers 99.73% of the multivariate normal process. The “modified tolerance region” is the largest ellipsoid centered at target vector with its axes parallel to the axes of the process ellipsoid and completely within the actual tolerance region. The direction of the process ellipsoid axes depends on the variance-covariance structure. So, they defined the first component as ${NMC}_{pm} = \frac{c^{'}}{\sqrt{χ_{p, 0.0027}^{2}}},$ (7)where $c^{'} = min {min {({USL}_{i} - T_{i}) / \sqrt{σ_{ii}}, (T_{i} - {LSL}_{i}) / \sqrt{σ_{ii}}}, i = 1, 2, . . ., p}$ , and σ_ii is variance for the characteristic i. The index NMC_pm is estimated as ${\hat{NMC}}_{pm} = \frac{\hat{c^{'}}}{\sqrt{χ_{p, 0.0027}^{2}}},$ (8)where $\hat{c^{'}} = min {min {({USL}_{i} - T_{i}) / \sqrt{s_{ii}}, (T_{i} - {LSL}_{i}) / \sqrt{s_{ii}}}, i = 1, 2, . . ., p}$ and s_ii is the sample variance for the ith characteristic.

If the value of ${\hat{NMC}}_{pm}$ exceeds 1, which indicates the region that covers 99.73% of the process region is smaller than the “modified tolerance region”, i.e. the part is “acceptable”. Otherwise, the process variability is more than allowable variability, so the process will be called incapable.

The second component of this vector is related to a hypothesis testing based on the assumption that the target is considered to be the true underlying mean of the process and it is the significant level of the observed value with the Hotelling’s T² statistic, $t^{2} = n (\bar{X} - T)^{'} S^{- 1} (\bar{X} - T)$ , expressed as $PV = P_{r} (F_{p, n - p} > \frac{n - p}{p (n - 1)} t^{2}),$ (9)where F_p,n-p denotes a random variable having the F distribution with p, n - p degrees of freedom. The PV value will never exceed 1 and the center of the process will be far from the engineering target vector when the PV value is close to zero.

The third component (LI), taking values, 0 or 1, compares the location of the region that covers 99.73% of the process with the tolerance region. It takes a value of 1 if the whole region that covers 99.73% of the process is contained within the tolerance region. Otherwise, it takes the value of 0. A process is called “capable” if the first component exceeds 1, the second component exceeds 0.05, and the third componentequals 1.

3 Fuzzy logic and basic definitions

Fuzzy logic is a form of a many-valued logic which deals with an approximating reasoning rather than a fixed and exact one. Compared to traditional binary sets (where variables may take on true or false values), fuzzy logic variables may have a true value that ranges from 0 to 1. Fuzzy logic has been extended to handle the concept of partial truth, where the true value may range from completely false to completely true [21]. Furthermore, when linguistic variables are used, these degrees may be managed by specific functions. This concept was introduced by Zadeh [42] as a means to model the uncertainty of natural language. Here we present some basic definitions.

Definition 1. (Fuzzy set) Suppose X is a nonempty set. A fuzzy set $\tilde{A}$ in X is characterized by its membership function $μ_{\tilde{A}} : X \to [0, 1]$ , $μ_{\tilde{A}} (x)$ is interpreted as the degree of membership of element x in fuzzy set $\tilde{A}$ for each x ∈ X. Also this function can be denoted by $\tilde{A}$ , that is $\tilde{A} : X \to [0, 1]$ . In this paper, we use the second notation.

Definition 2. (Support) Let $\tilde{A}$ be a fuzzy subset of X; the support of $\tilde{A}$ , denoted $supp (\tilde{A})$ , is a crisp subset of X whose elements all have nonzero membership grades in $\tilde{A}$ as the following $supp (\tilde{A}) = {x \in X ∣ \tilde{A} (x) > 0} .$

Definition 3. (Normal fuzzy set) A fuzzy subset $\tilde{A}$ of a classical set X is called normal if there exists an x ∈ X such that $\tilde{A} (x) = 1$ . Otherwise $\tilde{A}$ is subnormal.

Definition 4. (α-cut) An α-level set of a fuzzy set $\tilde{A}$ of X is a non-fuzzy set denoted by ${\tilde{A}}_{α}$ and is defined by ${\tilde{A}}_{α} = {\begin{matrix} {t \in X ∣ \tilde{A} (t) \geq α} & α > 0, \\ cl (supp (\tilde{A})) & α = 0 . \end{matrix}$ where $cl (supp (\tilde{A}))$ denotes the closure of the support of $\tilde{A}$ .

Definition 5. (Convex fuzzy set) A fuzzy set $\tilde{A}$ of X is called convex if ${\tilde{A}}_{α}$ is a convex subset of X,∀α ∈ [0, 1].

Definition 6. (Fuzzy number) A fuzzy number $\tilde{A}$ is a fuzzy set of the real line with a normal, (fuzzy) convex and continuous membership function of bounded support. The family of fuzzy numbers will be denoted by F.

Definition 7. Any fuzzy number $\tilde{A}$ can be described as $\tilde{A} (t) = {\begin{matrix} L (\frac{a - t}{α}) & t \in [a - α, a], \\ 1 & t \in [a, b], \\ R (\frac{t - b}{β}) & t \in [b, b + β], \\ 0 & otherwise . \end{matrix}$ where L : [0, 1] → [0, 1], R : [0, 1] → [0, 1], are continuous and non-increasing shape functions with L (0) = R (0) =1 and L (1) = R (1) =0. We call this fuzzy interval of LR-type and refer to it by $\tilde{A} = (a, b, α, β)_{LR}$ .

Definition 8. (Triangular fuzzy number) A fuzzy set $\tilde{A}$ is called triangular fuzzy number and is shown by T (a, b, c) if its membership function is as the following $\tilde{A} (x) = {\begin{matrix} (x - a) / (b - a) & a \leq x < b, \\ (c - x) / (c - b) & b \leq x < c, \\ 0 & otherwise . \end{matrix}$

Definition 9. (Trapezoidal fuzzy quantity) A fuzzy set $\tilde{A}$ is called trapezoidal fuzzy quantity and is shown by T_r (a, b, c, d) if its membership function is as the following $\tilde{A} (x) = {\begin{matrix} (x - a) / (b - a) & a \leq x < b, \\ 1 & b \leq x < c, \\ (d - x) / (d - c) & c \leq x < d, \\ 0 & otherwise . \end{matrix}$

4 Fuzzy multivariate process capability vector

As it was mentioned earlier, we consider a situation in which the specification limits and the target for each characteristic are not precise. In this section, we attend to two cases which the specification limits and target are triangular fuzzy numbers and trapezoidal fuzzy quantities (A fuzzy quantity has all properties of a fuzzy number with at least one normalelement).

4.1 Fuzzy specification limits and fuzzy target value

Suppose the lower and upper specification limitsare as $\tilde{LSL} = T (l_{1}, l_{2}, l_{3})$ and $\tilde{USL} = T (u_{1}, u_{2}, u_{3})$ , respectively. Moreover, the target is as $\tilde{T} = (t_{1}, t_{2}, t_{3})$ . So, their α-cut intervals are as thefollowing

$\begin{matrix} \tilde{USL} (α) & = & [U_{l} (α), U_{r} (α)] \\ = & [u_{1} + (u_{2} - u_{1}) α, u_{3} - (u_{3} - u_{2}) α], \\ \tilde{LSL} (α) & = & [L_{l} (α), L_{r} (α)] \\ = & [l_{1} + (l_{2} - l_{1}) α, l_{3} - (l_{3} - l_{2}) α], \\ \tilde{T} (α) & = & [T_{l} (α), T_{r} (α)] \\ = & [t_{1} + (t_{2} - t_{1}) α, t_{3} - (t_{3} - t_{2}) α] . \end{matrix}$ (10)

Now, suppose the specification limits and target are trapezoidal fuzzy quantity as $\tilde{LSL} = T_{r} (l_{1}, l_{2}, l_{3}, l_{4})$ , $\tilde{USL} = T_{r} (u_{1}, u_{2}, u_{3}, u_{4})$ and $\tilde{T} = T_{r} (t_{1}, t_{2}, t_{3}, t_{4})$ . Their α-cut intervals are as the following $\begin{matrix} \tilde{USL} (α) & = & [U_{l} (α), U_{r} (α)] \\ = & [u_{1} + (u_{2} - u_{1}) α, u_{4} - (u_{4} - u_{3}) α], \\ \tilde{LSL} (α) & = & [L_{l} (α), L_{r} (α)] \\ = & [l_{1} + (l_{2} - l_{1}) α, l_{4} - (l_{4} - l_{3}) α], \\ \tilde{T} (α) & = & [T_{l} (α), T_{r} (α)] \\ = & [t_{1} + (t_{2} - t_{1}) α, t_{4} - (t_{4} - t_{3}) α] . \end{matrix}$ (11)

4.2 Ranking function

To make a comparison between two fuzzy numbers (quantities), we employ the ranking method proposed by Fortemps and Roubens [6]. If $\tilde{A}$ and $\tilde{B}$ are two fuzzy numbers (quantities), we will then have the following relations;

$\begin{matrix} \tilde{A} & \geq_{R} & \tilde{B} iff R (\tilde{A}) \geq R (\tilde{B}), \\ \tilde{A} & >_{R} & \tilde{B} iff R (\tilde{A}) > R (\tilde{B}), \\ \tilde{A} & =_{R} & \tilde{B} iff R (\tilde{A}) = R (\tilde{B}), \end{matrix}$ (12)where the definition of $R (\tilde{W})$ is given as what follows; $R (\tilde{W}) = \frac{1}{2} \int_{0}^{1} ({\tilde{W}}_{l} (α) + {\tilde{W}}_{r} (α)) d α,$ (13)and ${\tilde{W}}_{l} (α)$ and ${\tilde{W}}_{r} (α)$ are the lower and the upper limits of the α-cut intervals. So, if $\tilde{A} \geq_{R} \tilde{B}$ , then $min {\tilde{A}, \tilde{B}} = \tilde{B}$ .

4.3 Fuzzy estimation of the index MC_pm

Here we introduce ${\tilde{MC}}_{pm}$ which is as ${\tilde{MC}}_{pm} = \frac{{\tilde{MC}}_{p}}{\tilde{D}} .$ (14)

To compute ${\tilde{MC}}_{p}$ , we should obtain the fuzzy radiuses of the modified tolerance region ellipsoid as the following

${\tilde{r}}_{i} = min {{\tilde{USL}}_{i} ⊖ {\tilde{T}}_{i}, {\tilde{T}}_{i} ⊖ {\tilde{LSL}}_{i}},$ (15)for i = 1, 2, . . . , p.

To find ${\tilde{r}}_{i}$ , we apply ranking function mentioned earlier. Then, we can compute volume of the ellipsoid. For example, for the process with two characteristic, modified tolerance region is an ellipse and the numerator of ${\tilde{MC}}_{p}$ is calculated as follows

$vol . ({\tilde{R}}_{1}) = π \otimes {\tilde{r}}_{1} \otimes {\tilde{r}}_{2} .$

It would be preferable to obtain the α-cut intervals of above factor. Since ${\tilde{r}}_{1} (α) = [r_{1 l} (α), r_{1 r} (α)]$ and ${\tilde{r}}_{2} (α) = [r_{2 l} (α), r_{2 r} (α)]$ , we have

$vol . ({\tilde{R}}_{1}) (α) = [π r_{1 l} (α) r_{2 l} (α), π r_{1 r} (α) r_{2 r} (α)] .$

Therefore, the α-cut intervals of ${\tilde{MC}}_{p}$ is as

$\begin{matrix} {\tilde{MC}}_{p} (α) & = & [{MC}_{pl} (α), {MC}_{pr} (α)] \\ = & [\frac{r_{1 l} (α) r_{2 l} (α)}{∣ S ∣^{1 / 2} χ_{2, 0.0027}^{2}}, \frac{r_{1 r} (α) r_{2 r} (α)}{∣ S ∣^{1 / 2} χ_{2, 0.0027}^{2}}] . \end{matrix}$

For more characteristic, we do similarly.

Now, we should fuzzify $\hat{D}$ . We obtain the α-cut intervals of $\tilde{\hat{D}}$ as follows $\tilde{\hat{D}} (α) = [{\hat{D}}_{l} (α), {\hat{D}}_{r} (α)],$ (16)where

$\begin{matrix} {\hat{D}}_{l} (α) & = & min {[1 + \frac{n}{n - 1} {(\begin{matrix} {\bar{X}}_{1} - t_{1} \\ {\bar{X}}_{2} - t_{2} \\ ⋮ \\ {\bar{X}}_{p} - t_{p} \end{matrix})}^{'} S^{- 1} \\ (\begin{matrix} {\bar{X}}_{1} - t_{1} \\ {\bar{X}}_{2} - t_{2} \\ ⋮ \\ {\bar{X}}_{p} - t_{p} \end{matrix})]^{1 / 2}; t_{i} \in T_{ij} (α), \\ i & = & {1, 2, . . ., p}, j = {l, r}}, \end{matrix}$ (17)and

$\begin{matrix} {\hat{D}}_{r} (α) & = & max {[1 + \frac{n}{n - 1} {(\begin{matrix} {\bar{X}}_{1} - t_{1} \\ {\bar{X}}_{2} - t_{2} \\ ⋮ \\ {\bar{X}}_{p} - t_{p} \end{matrix})}^{'} S^{- 1} \\ (\begin{matrix} {\bar{X}}_{1} - t_{1} \\ {\bar{X}}_{2} - t_{2} \\ ⋮ \\ {\bar{X}}_{p} - t_{p} \end{matrix})]^{1 / 2}; t_{i} \in T_{ij} (α), \\ i & = & {1, 2, . . ., p}, j = {l, r}} . \end{matrix}$ (18)

Therefore, the α-cut intervals of ${\tilde{MC}}_{pm}$ is as the following

$\begin{matrix} {\tilde{MC}}_{pm} (α) & = & [{MC}_{pml} (α), {MC}_{pmr} (α)] \\ = & [\frac{{MC}_{pl} (α)}{{\hat{D}}_{r} (α)}, \frac{{MC}_{pr} (α)}{{\hat{D}}_{l} (α)}] . \end{matrix}$ (19)

Now we place these α-cut intervals, one on top of the other to produce ${\tilde{MC}}_{pm}$ . The process would be capable if ${\tilde{MC}}_{pm}$ is at least approximately (approx.) one. To make a decision, we should compare this index with the fuzzy number 1, i.e. by using ranking function mentioned in the previous subsection, we determine the minimum of the fuzzy index and 1. If the minimum is 1, then we conclude that the process is capable.

4.4 Fuzzy estimation of the index NMPCV

In this subsection, we introduce $N \tilde{MPC} V = [{\tilde{NMC}}_{pm}, \tilde{PV}, LI]$ . The first component is as the following ${\tilde{NMC}}_{pm} = \frac{\tilde{c^{'}}}{\sqrt{χ_{p, 0.0027}^{2}}},$ (20)where

$\begin{matrix} \tilde{c^{'}} = min {min & { & \frac{{\tilde{USL}}_{i} ⊖ {\tilde{T}}_{i}}{{\sqrt{σ}}_{ii}}, \frac{{\tilde{T}}_{i} ⊖ {\tilde{LSL}}_{i}}{{\sqrt{σ}}_{ii}}}; \\ i & = 1, 2, . . ., p} . \end{matrix}$ (21)

Based on the given random sample, we can estimate this index as ${\tilde{\hat{NMC}}}_{pm} = \frac{\tilde{\hat{c^{'}}}}{\sqrt{χ_{p, 0.0027}^{2}}},$ (22)where

$\begin{matrix} \tilde{\hat{c^{'}}} = min & {min { & \frac{{\tilde{USL}}_{i} ⊖ {\tilde{T}}_{i}}{{\sqrt{s}}_{ii}}, \frac{{\tilde{T}}_{i} ⊖ {\tilde{LSL}}_{i}}{{\sqrt{s}}_{ii}}}; \\ i = 1, 2, . . ., p} . \end{matrix}$ (23)

To get $\tilde{\hat{c^{'}}}$ , first we obtain α-cut intervals of each item in above equation as follows

$\begin{matrix} \frac{{\tilde{USl}}_{i} ⊖ {\tilde{T}}_{i}}{\sqrt{s_{ii}}} (α) & = & (\frac{u_{l} (α) - t_{r} (α)}{\sqrt{s_{ii}}}, \frac{u_{r} (α) - t_{l} (α)}{\sqrt{s_{ii}}}), \\ \frac{{\tilde{T}}_{i} ⊖ {\tilde{LSL}}_{i}}{\sqrt{s_{ii}}} (α) & = & (\frac{t_{l} (α) - l_{r} (α)}{\sqrt{s_{ii}}}, \frac{t_{r} (α) - l_{l} (α)}{\sqrt{s_{ii}}}) . \end{matrix}$ (24)

Then, we apply ranking function and construct the fuzzy number (quantity) $\tilde{\hat{c^{'}}}$ by its α-cut intervals and so we can obtain ${\tilde{\hat{NMC}}}_{pm}$ . To make a decision, we should compare this index with the fuzzy number 1, i.e., by using ranking function mentioned in the previous subsection, we determine the minimum of the fuzzy index and 1. If the minimum is 1, then we conclude that the spread of the process is less than the tolerance spread.

To gain $\tilde{PV}$ , we get the α-cut intervals as thefollowing

$\begin{matrix} {PV}_{l} (α) = min {P_{r} (F_{p, n - p} > \frac{n - p}{p (n - 1)} \\ {(\begin{matrix} {\bar{X}}_{1} - t_{1} \\ {\bar{X}}_{2} - t_{2} \\ ⋮ \\ {\bar{X}}_{p} - t_{p} \end{matrix})}^{'} S^{- 1} (\begin{matrix} {\bar{X}}_{1} - t_{1} \\ {\bar{X}}_{2} - t_{2} \\ ⋮ \\ {\bar{X}}_{p} - t_{p} \end{matrix})); \\ t_{i} \in T_{ij} (α), i = {1, 2, . . ., p}, j = {l, r}}, \end{matrix}$ (25)and

$\begin{matrix} {PV}_{r} (α) = max {P_{r} (F_{p, n - p} > \frac{n - p}{p (n - 1)} \\ {(\begin{matrix} {\bar{X}}_{1} - t_{1} \\ {\bar{X}}_{2} - t_{2} \\ ⋮ \\ {\bar{X}}_{p} - t_{p} \end{matrix})}^{'} S^{- 1} (\begin{matrix} {\bar{X}}_{1} - t_{1} \\ {\bar{X}}_{2} - t_{2} \\ ⋮ \\ {\bar{X}}_{p} - t_{p} \end{matrix})); \\ t_{i} \in T_{ij} (α), i = {1, 2, . . ., p}, j = {l, r}} . \end{matrix}$ (26)

Several answers are obtained based on above equations, but we choose the ones satisfy in the following condition:

$\begin{matrix} {PV}_{l} (α) & \leq & {PV}_{l} (β) \leq {PV}_{l} (1) = {PV}_{r} (1) \\ \leq & {PV}_{r} (β) \leq {PV}_{r} (α); \\ \forall & α < β, α, β \in [0, 1] . \end{matrix}$ (27)

Hence, we can construct the fuzzy p-value by its α-cut intervals. To make a decision, first we detect a degree of uncertainty such γ (between 0 and 1). Then, we have a three decision problem as what follows;

If PV_l (γ) >0.05, then the process mean is not far from the target.

If PV_r (γ) <0.05, then the process mean is far from the target.

If PV_l (γ) ≤0.05 ≤ PV_r (γ), then we cannot come to a clear decision. In such cases, one may take more samples and follow the procedure until making a decision.

For the third component, we defuzzify the specification limits and target. So, by the geometry of tolerance and process ellipses, we can obtain this component. For defuzzification, we use ranking function R as mentioned in subsection 4.2.

5 Numerical examples

In this section, we apply the proposed indices for two examples in which the first one is related to a process with triangular fuzzy numbers specification limits and targets, the second one relates to a process with trapezoidal fuzzy quantity specification limits andtargets.

5.1 Example 1

Jackson [10] discussed a film-developing solution process with two quality characteristics, Elon and Hydroguinone. Suppose specification limits and target are triangular fuzzy numbers as shown in Table 1.

Seventy-five (crisp vectors) samples were taken and the sample mean vector and sample variance-covariance matrix are as follows

$\bar{X} = (\begin{matrix} 264.32 \\ 471.48 \end{matrix}), S = (\begin{matrix} 102.65 & 68.87 \\ 68.87 & 107.96 \end{matrix})$

We obtain ${\tilde{MC}}_{p}$ as “approximately 0.9556” and then get ${\tilde{MC}}_{pm}$ as “approximately 0.9255”, in fact ${\tilde{MC}}_{pm} = T (0.4741, 0.9255, 1.0861)$ . Figure 1 shows the membership function of this index. Based on ranking function, we get $R ({\tilde{MC}}_{pm}) = 0.92105$ , so ${\tilde{MC}}_{pm} <_{R} \tilde{1}$ and we conclude the process is non-capable.

Also, the estimation of $\tilde{c^{'}}$ is a triangular fuzzy number as $T (28 / \sqrt{107.96}, 30 / \sqrt{107.96}, 32 / \sqrt{107.96})$ . Then, the first component of fuzzy multivariate capability vector is obtained “approximately 0.8395” as

${\tilde{\hat{NMC}}}_{pm} = T (0.784, 0.8395, 0.895) .$

Figure 2 shows the membership function of this index. To make a decision, we obtain $R ({\tilde{\hat{NMC}}}_{pm}) = 0.8395$ . So, we have ${\tilde{\hat{NMC}}}_{pm} <_{R} \tilde{1}$ .

$\tilde{PV}$ is obtained “approximately 0.9684”. Based on the α-cut intervals of $\tilde{PV}$ , we obtain the membership function of fuzzy p-value as shown in Fig. 3. To make a decision, we consider the degree of uncertainty as 0.75, so, we get $\tilde{PV} (0.75) = [0.9528, 0.9809]$ . It is obvious that PV_l (0.75) >0.05, so the process mean vector is not far from the target vector.

To defuzzify the specification limits and target for two characteristics, we applied the ranking function and results shown in Table 2. Figure 4 displays these crisp specification limits and target and 99.73% of process region. Intuitively, we can see that the region which covers 99.73% of process exceeds the tolerance region, so, LI = 0. Hence, the process is incapable because of unallowable variability.

5.2 Example 2

Sultan [35] discussed a raw material process which its quality is affected by two characteristics, Brinell hardness and Tensile strength. Let the trapezoidal fuzzy quantity specification limits and targets for these characteristics are as indicated in Table 3.

The data consisted of 25 observations. The sample mean vector was $\bar{X} = [177.2, 52.32]^{'}$ and the sample variance-covariance matrix was

$S = (\begin{matrix} 337.8 & 58.3308 \\ 58.3308 & 33.6247 \end{matrix}) .$

Therefore, the estimation of ${\tilde{MC}}_{p}$ is a trapezoidal fuzzy quantity “approximately between 1.5631 and 1.8992” and so the estimation of ${\tilde{MC}}_{pm}$ is as “approximately between 1.4867 and 1.8874” which is shown in Fig. 5. We obtain $R ({\tilde{MC}}_{pm}) = 1.67915$ which exceeds one. So, the process is capable.

Furthermore, the estimation of $\tilde{c^{'}}$ is a trapezoidal fuzzy quantity and then, the first component of fuzzy multivariate capability vector is gotten “approximately between 0.9935 and 1.0409” and its membership function is as shown in Fig. 6.

To make a decision, we get $R ({\tilde{\hat{NMC}}}_{pm}) = 1.0172$ , it is bigger than 1. So, we have ${\tilde{\hat{NMC}}}_{pm} >_{R} \tilde{1}$ . Hence, the process variability is approximately less than the allowable variability.

$\tilde{PV}$ is obtained “approximately between 0.9530 and 0.9943”. Figure 7 shows the membership function of fuzzy p-value. To make a decision, we consider the degree of uncertainty as 0.75. So, we get $\tilde{PV} (0.75) = [0.9949, 0.9963]$ . It is obvious that PV_l (0.75) >0.05, so the process mean vector is not far from the target vector.

Table 4 shows defizified specification limits and target. The modified tolerance region and the region which covers 99.73% of the process are shown in Fig. 8.

Figure 8 shows that the region which covers 99.73% of the process, slightly exceeds the tolerance region. So, the third component is as LI = 0. Hence, the process is incapable.

Table 5 shows a comparison between two proposed indices for examples listed. Based on the results, we can conclude that fuzzy estimation of NMPCV works better than fuzzy estimation of MC_pm. We note that Wang et al. [39] show NMPCV works better than MC_pm in crisp situations.

6 Conclusion

In this paper, we reviewed two useful multivariate process capability index, first. Then, we referred to circumstances in which the specification limits and the target are not precise, so, they can be explained as fuzzy numbers (quantities). We introduced fuzzy multivariate capability indices for two cases whose specification limits and the target are triangular fuzzy numbers and trapezoidal fuzzy quantities. Finally, we applied our new fuzzy multivariate capability indices for two numerical examples to demonstrate the performance of the new indicies. We concluded that one of them is superior to another one as it is in crisp situation.

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