Indices for Multivariate Processes

In this subsection, consider a multivariate process with k quality characteristics. Sup-pose X₁, . . . , X_n are n_i i.i.d. k × 1 random vectors of observations. ¯X is a k × 1 vector representing the sample mean of X₁, . . . , X_n.

Assuming the process data X follows a multivariate normal distribution with mean µ and variance-covariance matrix Σ, denoted by X ∼ N_k(µ, Σ). Chan et al. [6] proposed an index for measuring how far the process mean µ is from the target value T as

Cepm =

µ k

E[(X − T )^TΣ⁻¹(X − T )]

¶_1/2 .

Pearn et al. [16] introduced two multivariate PCIs, which are viewed as more nat-ural generalizations of Cpm than the one proposed by Chan et al. [6]. They defined a multivariate C_p index as

kC_p² = c² χ²_k,0.0027,

where χ²_k,α is the upper α quantile of a chi-square distribution with degrees of freedom k, and c is a constant satisfying P {(X − T )^TΣ⁻¹(X − T ) ≤ c²} = 0.9973. Analogously, they defined a multivariate C_pm index by

kC_pm² = ^kC_p²

1 + (µ − T )^TΣ⁻¹(µ − T )/k. (1) Hubele et al. [13] proposed an index vector (C_{P M}, P V, LI)^T for bivariate normal pro-cesses. The first component

C_{P M} = area of specification

area of modified process region =



The modified process region is the smallest rectangle that can circumscribe 100(1−α)% of the process distribution (see Fig 1). The edges of the modified process region are defined as the lower and upper process limits, LP L_i and UP L_i, i = 1, 2. These four values can be obtained by solving the system of equations of first derivatives with respect to each xi

of

(X − µ)^TΣ⁻¹(X − µ) = χ²_{2, α}, where X = (X1, X2)^T and µ = (µ1, µ2)^T.

The solutions are

UP L1 = µ1+ s

χ²_2,αdet(Σ⁻¹₁ )

det(Σ⁻¹) , LP L1 = µ1− s

χ²_2,αdet(Σ⁻¹₁ ) det(Σ⁻¹) , UP L₂ = µ₂+

s

χ²_2,αdet(Σ⁻¹₂ )

det(Σ⁻¹) , LP L₂ = µ₂− s

χ²_2,αdet(Σ⁻¹₂ ) det(Σ⁻¹) ,

where Σ⁻¹_i , i = 1, 2, is the matrix obtained from Σ⁻¹ by deleting the ith row and column.

The meaning of this component is analogous to that of C_p, measuring the variation of product characteristics relative to the specifications.

Figure 1: Explaining diagram of C_{P M}

The second component is the p-value of testing the difference between the center of specification (target value T ) and the process mean. Let the null hypothesis H₀ : µ = T , the Hotelling T² statistic [11] is

T² = n( ¯X − T )^TΣˆ⁻¹( ¯X − T ),

where ¯X is the sample mean and ˆΣ is the usual sample variance-covariance matrix of process data. Since n − 2

2(n − 1)T² follows F2,n−2 distribution under null hypothesis, the p-value-based component P V is defined as

P V = P

³

T² ≥ 2(n − 1)

n − 2 F^2,n−2,α

´ ,

where F_2,n−2,α stands for the 100(1 − α)% percentile of F distribution with degrees of freedom 2 and n − 2. This component measures the distance of the process mean and the

target value. If the process mean is close to the target value, P V will be close to 1.

The third component LI provides the information about the location of the modified process region relative to the specification, defined as

LI = max

³

1, ^{|U P L}_{U SL}¹₁^{−U SL}_−LSL1^1|, ^{|LP L}_{U SL}¹₁^−LSL_−LSL¹₁^|, ^{|U P L}_{U SL}²₂^{−U SL}_−LSL2^2|, ^{|LP L}_{U SL}²₂^−LSL_−LSL²₂^|

´ .

If this component is equal to 1, then the entire modified process region falls within or on the specification. If the component is greater than 1, then some or all modified process region falls out of the specification.

This index vector contains three components summarizing the size and location of process contour related to the specification.

Taam et al. [18] proposed an index as the ratio of two areas

Cf_p = Area(R₁)

Area(R₂) = Area(modified specification)

Area(99.73% process region), (2)

Figure 2: Explaining diagram of fC_p

where R₁ is a modified specification, which is the largest ellipsoid that is centered at the target value and completely within the original specification, R2 is an elliptical region containing 99.73% of the bivariate normal distribution. This index is an extension of the univariate Cp for bivariate processes. Considering the shift of process mean from the

target value T , Taam et al. [18] further modified this index by taking an adjustment factor D into account and defined a C_pm index for two quality characteristics as follows:

MC_pm = Cf_p

0 < D⁻¹ < 1 measures the closeness between the process mean and the target value. A larger value of 0 < D⁻¹ < 1 implies that the process mean is closer to the target value.

Chen [7] proposed an index using the concept of a specification zone expressed as V (r₀) = {x ∈ R^k: h(x − µ₀) ≤ r₀}, (3) where h(·) is a nonnegative homogeneous scalar function satisfying the condition h(tx) = th(x) for all t > 0 and r₀ is a positive number. A process is considered capable if P (X ∈ V (r₀)) ≥ 1 − α, where α is the allowable expected proportion of non-conforming products. Let r = min{c : P (X ∈ V (c)) ≥ 1 − α}. Then a process is considered to be capable if and only if r ≤ r₀. This leads one to express an index for multivariate process in the form

MC_p = r₀ r .

According to Chen [7], this definition provides the following advantages: (i) allowing flexible specifications as general as given by V (r0) in (3), (ii) assuming no conditions on the underlying distribution, and (iii) permiting flexibility in setting a criterion for the capability of a process. For example, consider a rectangular specification zone

W = {x ∈ R^k: |x_i− µ_i| ≤ r_i, i = 1, . . . , k},

where µ is the process mean and r_i’s are positive constants. One can derive an alternative definition of MC_p as

MC_p = 1 r^∗, where r^∗ is a constant satisfying P

³

max{|X_i− µ_i|/r_i, i = 1, . . . , k} ≤ r^∗

´

= 1 − α. If MCp ≥ 1, the process is capable at 100(1 − α)% confidence level.

Pal [15] proposed an index defined as follows:

C_{P B} = SR

A_p = (USL1− LSL1)(USL2− LSL2) πχ²_2,0.0027p

σ₁²σ₂²− σ²₁₂ ,

where S_R represents the area of the specification rectangle and A_p represents the 99.73%

area of the process region. This index is, in fact, an extension of the index (2) proposed by Taam et al. [18]. It is an area ratio of a rectangular region over an elliptical region while Taam et al. [18] used an elliptical region over another elliptical region as the area ratio.

Bothe [2] proposed a multivariate C_pk index defined as MCpk = ZPT

3 ,

where Z_PT is the P_Tth percentile of the standard normal distribution, and P_T is defined as

P_T = 1 − ((1 − P_QC1)(1 − P_QC2) · · · (1 − P_QCk))^k1

with P_QCi, i = 1, . . . , k, being the non-conforming rate of the ith quality characteristic.

However, this index is designed only for independent process characteristics.

Wang and Du [19] proposed a method using principal component (PC) analysis to describe the performance of a process of multiple characteristics. In that paper, the pro-cedures of obtaining the indices for normal data as well as non-normal data are described in the following:

and the elements in the above expressions are given in the following.

Suppose S is a non-singular k × k sample variance-covariance matrix. LSL and U SL are k × 1 vectors of lower and upper specification limits, respectively. Using spectral de-composition, we can obtain a matrix D = U^TSU , where D = diag(λ²₁, λ²₂, . . . , λ²_k) with λ²₁ ≥

λ²₂ ≥ · · · ≥ λ²_k being the eigenvalues of S, and the columns of U , u₁, u₂, . . . , u_k, are the associated eigenvectors. As a result,

S_{P Ci} = λ_i, ¯X_{P Ci} = u^T_iX,¯

USL_{P Ci} = u^T_i U SL, LSL_{P Ci} = u^T_i LSL, i = 1, . . . , k, d = 1k¯ 1

n Pk i=1

Pn j=1

¯¯

¯u_i^TX_j− USL_{P Ci}+ LSL_{P Ci} 2

¯¯

¯ .

Here we remark that the numerators of dMCp and dMCpk seem somewhat unreasonable, since the vectors U SL and LSL no longer represent upper or lower bounds of the spec-ification region in the directions of principal components. As a result, USLP Ci− LSLP Ci

sometimes may even become negative.

Wang et al. [20] compared three process capability indices: (C_{P M}, P V, LI)^T pro-posed by Hubele et al. [13], MC_pm proposed by Taam et al. [18], and MC_p proposed by Chen [7]. They summarized that, in general, the multivariate indices could be obtained from (i) the area ratio of a specification region to a process region, (ii) the probability of a non-conforming product, and (iii) other approaches using loss functions or vector representation. The purpose of Wang et al. [20] is to illustrate the distinctions among the various meanings of capability in the multivariate case.

The purpose of this paper is to study yield related PCIs for multivariate processes.

As mentioned in Section 1, BC_pk and BC_p proposed by Castagliola and Castellanos [4]

are such indices. We shall give a more detailed review on these indices and then present how we would extend BC_pk to higher dimensions and how to modify BC_p to become scale-invariant in the later sections. And the last but not the least, we will provide methodologies on how to compute these indices.

3 Multivariate C

_pk

Index

3.1 Yield Measuring Index for Processes with Multiple Charac-teristics

In this subsection, we first introduce the bivariate C_pk index, BC_pk, proposed by Castagliola and Castellanos [4]. Then we provide the link between BC_pk and yield. More-over, we extend this index to higher dimensions.

3.1.1 Alternative Definition of C_pk

Assume that the quality characteristic X of a product item is a N(µ, σ²) random vari-able. Let [LSL, USL] be the corresponding lower and upper specification limits. Equiv-alent to the definition of Kane [14], an alternative definition for Cpk was proposed by Castagliola and Castellanos [4]. This definition is based on the lower and upper propor-tions of non-conforming products pL = P (X ≤ LSL) and pU = P (X ≥ USL). Since X

∼ N(µ, σ²), p_L = Φ(^LSL−µ_σ ) and p_U = Φ(^{−U SL+µ}_σ ), where Φ is the cumulative distribution function (c.d.f.) of the standard normal distribution. Moreover, since the cumulative distribution function Φ is a strictly increasing function of the random variable, C_pk is equivalent to

1

3min{−Φ⁻¹(p_U), −Φ⁻¹(p_L)}. (4) Similarly, the C_p in Kane [14] is equivalent to

1 6

³

−Φ⁻¹(pU) − Φ⁻¹(pL)

´ .

3.1.2 Definition of BC_pk

Let X1 and X2 be the quality characteristics of interests with the specification limits [LSL₁, USL₁] for X₁ and [LSL₂, U SL₂] for X₂. These limits define a rectangular specifi-cation area A. Assume that X = (X1, X2)^T follows a bivariate normal distribution with mean µ = (µ₁, µ₂)^T and variance-covariance matrix Σ. Applying eigenvalue-eigenvector decomposition to Σ, we obtain two eigenvalues λ²₁ ≥ λ²₂ > 0 and the associated eigenvec-tors, v₁ and v₂. Let R = [v₁, v₂], then R^TR = I and Σ can be expressed as Σ = RV R^T,

where V is the diagonal matrix with diagonal elements λ²₁ and λ²₂. In fact, the matrix R represents the rotation matrix that rotates the original axes to the main axes of the bivariate normal distribution (see Figure 3), v₁ and v₂ correspond to the main axes, and λ²₁ and λ²₂ are the variances on these main axes. More specifically, if we let S_i = v_i^TX, then S_i ∼ N(v_i^Tµ, λ²_i), i = 1, 2, and S₁ and S₂ are independent. Suppose we move the origin to the process mean µ and have the two new axes being in the directions of v₁ and v₂. Then the two main axes divide the plane into four regions, A₁, A₂, A₃, and A₄. Obviously, P (X ∈ A_i) = 1/4, i = 1, . . . , 4. Denoting the specification region by A and Q_i = A_i∩ A, i = 1, . . . , 4. Let q_i = P (X ∈ Q_i), i = 1, . . . , 4. Then the probability that X is in A_i but not in the specification region is p_i = 1/4 − q_i (see Figure 3).

Figure 3: Explaning diagram of BC_pk

By analogy to the alternative definition of C_pk given in (4), Castagliola and Castel-lanos [4] defined a bivariate Cpk as

BC_pk = 1

3min(−Φ⁻¹(2p₁), −Φ⁻¹(2p₂), −Φ⁻¹(2p₃), −Φ⁻¹(2p₄)).

This definition is similar to the alternative definition of Cpk, except that 0 ≤ pi ≤ 1/4, i = 1, . . . , 4, in the bivariate case, while 0 ≤ p_u(or p_L) ≤ 1/2 in the univarite case. We extend this definition to higher dimensions later.

3.1.3 Non-conforming Rate Based on BC_pk

According to the definition of BC_pkin the last subsection, we can establish a connection between the non-conforming rate (%NC) and BC_pk. First note that

BC_pk = 1

3min(−Φ⁻¹(2p₁), −Φ⁻¹(2p₂), −Φ⁻¹(2p₃), −Φ⁻¹(2p₄))

= −1

3max(Φ⁻¹(2p₁), Φ⁻¹(2p₂), Φ⁻¹(2p₃), Φ⁻¹(2p₄)).

Since Φ⁻¹(·) is a strictly increasing function, BC_pk = −1

3Φ⁻¹(2p_max),

where p_max = max(p₁, p₂, p₃, p₄). Φ⁻¹(·) is a one-to-one function, so p_max = 1

2Φ(−3BC_pk). (5)

Note that pmax ≤ %NC ≤ 4pmax. Plugging (5) into this inequality, we obtain 1

2Φ(−3BCpk) ≤ %NC ≤ 2Φ(−3BCpk). (6) Although the lower bound of (6) is quite conservative, it is a convenient bound, mean-ing once the engineer gets a BC_pk value, he/she will know the bounds of non-conforming rate. The upper bound is very useful and is not a loose bound, meaning that it is reachable. Usually producers can take the upper bound of the non-conforming rate as an quality assurance to customers. For example, if the process is with BC_pk=1.00, one can guarantee that there will be 2700 non-conformities in 1,000,000 product items at most.

Table 1 gives the upper and lower bounds of the non-conforming rate %NC for various values of BC_pk. Figure 4 plots the bounds. We can see the bounds drop sharply as BC_pk increases and soon levels off when BC_pk ≥ 1.33.

The second inequality of (6) is equivalent to

2Φ(3BC_pk) − 1 ≤ % yield,

providing a same lower bound for the yield as in the univariate case [3]. The lower bound gives the worst level of the yield for a given BC_pk.

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

010000300005000070000

BCpk

NCppm

upb lwb

Figure 4: Bounds of non-conformity based on BC_pk

Table 1: Bounds of non-conformity based on BC_pk BC_pk Non-conformities in ppm

lwb upb

0.60 17965.15956 71860.63823 0.80 4098.76796 16395.07185 1.00 674.94902 2699.79606 1.33 16.51832 66.07330

1.50 1.69884 6.79535

1.60 0.39666 1.58666

1.67 0.13608 0.54430

2.00 0.00049 0.00197

3.1.4 Extending C_pk to Higher Dimensions

Now we generalize the alternative definitions of C_pkand BC_pkto multivariate processes of k > 2 characteristics. By the same notion for the bivariate case, dividing the space R^k into 2^k subregions by the main axes of the k-variate distribution, we can define a multivariate C_pk index as

MC_pk = 1

3min(−Φ⁻¹(2^k−1p₁), −Φ⁻¹(2^k−1p₂), . . . , −Φ⁻¹(2^k−1p₂^k))

= −1

3max(Φ⁻¹(2^k−1p₁), Φ⁻¹(2^k−1p₂), . . . , Φ⁻¹(2^k−1p₂^k))

= −1

3Φ⁻¹(2^k−1p_max),

where p_i is the probability of a randomly selected sample being in the ith subregion, but not meeting the specification and p_max = max(p₁, p₂, . . . , p₂^k). Equivalently,

p_max= 1

2^k−1Φ(−3MC_pk).

Since p_max ≤ %NC ≤ 2^kp_max, we can also get an inequality of non-conforming rate in the general multivariate case as

1

2^k−1Φ(−3MC_pk) ≤ %NC ≤ 2Φ(−3MC_pk), which is equivalent to

1 − 1

2^k−1Φ(−3MC_pk) ≤ %yield ≤ 1 − 2Φ(−3MC_pk), (7)

在文檔中 多變量製程能力指標之研究 (頁 12-22)