7.4 Special Process Control & Advanced SPC

7.4 Special Process Control & Advanced SPC

Traditional Shewhart control charts ($\bar{X}-R$, $\bar{X}-S$, and $I-MR$) rely exclusively on the information contained in the most recent sample point. While highly effective for detecting large process shifts (greater than $1.5\sigma$ to $2.0\sigma$), Shewhart charts possess a "memoryless" property that makes them relatively insensitive to small, persistent process shifts ($0.5\sigma$ to $1.5\sigma$). Furthermore, modern manufacturing environments frequently encounter low-volume high-mix production (short runs) or multiple correlated quality attributes that violate the assumptions of univariate Shewhart control charts.

Advanced Statistical Process Control (SPC) techniques—including Exponentially Weighted Moving Average (EWMA), Cumulative Sum (CUSUM), Short-Run SPC, and Multivariate SPC ($Hotelling's\ T^2$)—provide quality engineers with specialized tools to maintain statistical control under these challenging conditions.

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1. Exponentially Weighted Moving Average (EWMA) Charts

The Exponentially Weighted Moving Average (EWMA) control chart monitors small process shifts by calculating a moving average that assigns geometrically decreasing weights to older observations.

Mathematical Formulation

The EWMA statistic $z_i$ is calculated recursively:

$$z_i = \lambda x_i + (1 - \lambda) z_{i-1}$$

Where:

• $x_i$ is the $i$-th sample observation (or sample mean $\bar{X}_i$).
• $\lambda$ is the weighting constant, constrained to $0 < \lambda \le 1$.
• $z_0 = \mu_0$ (the target or historical process mean).

When $\lambda = 1$, the EWMA chart simplifies to a standard Shewhart individual control chart. As $\lambda \to 0$, past observations exert greater influence, increasing sensitivity to small shifts.

Control Limits Calculation

The variance of the EWMA statistic $z_i$ is given by:

$$\sigma_{z_i}^2 = \sigma^2 \left( \frac{\lambda}{2 - \lambda} \right) \left[ 1 - (1 - \lambda)^{2i} \right]$$

For individual observations ($n=1$), the upper and lower control limits are:

$$UCL = \mu_0 + L \sigma \sqrt{\frac{\lambda}{2 - \lambda} \left[ 1 - (1 - \lambda)^{2i} \right]}$$ $$LCL = \mu_0 - L \sigma \sqrt{\frac{\lambda}{2 - \lambda} \left[ 1 - (1 - \lambda)^{2i} \right]}$$

Where $L$ is the width factor of the control limits (typically $L \approx 2.7$ to $3.0$). As the sample index $i$ increases, the term $(1 - \lambda)^{2i} \to 0$, yielding the steady-state asymptotic control limits:

$$UCL_{\text{steady}} = \mu_0 + L \sigma \sqrt{\frac{\lambda}{2 - \lambda}}$$ $$LCL_{\text{steady}} = \mu_0 - L \sigma \sqrt{\frac{\lambda}{2 - \lambda}}$$

Worked Example: EWMA Calculation

Suppose a chemical process has a target viscosity $\mu_0 = 50.0 \text{ cP}$ and known standard deviation $\sigma = 2.0 \text{ cP}$. We select $\lambda = 0.10$ and $L = 2.7$.

1. Steady-state control limits: $$UCL = 50.0 + 2.7 (2.0) \sqrt{\frac{0.10}{2 - 0.10}} = 50.0 + 5.4 \sqrt{\frac{0.10}{1.90}} = 50.0 + 5.4 (0.2294) = 51.24 \text{ cP}$$ $$LCL = 50.0 - 5.4 (0.2294) = 48.76 \text{ cP}$$
2. First observation ($x_1 = 52.5 \text{ cP}$): $$z_1 = 0.10 (52.5) + 0.90 (50.0) = 5.25 + 45.0 = 50.25 \text{ cP}$$
3. Second observation ($x_2 = 53.0 \text{ cP}$): $$z_2 = 0.10 (53.0) + 0.90 (50.25) = 5.30 + 45.225 = 50.525 \text{ cP}$$

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2. Cumulative Sum (CUSUM) Control Charts

The Cumulative Sum (CUSUM) chart directly accumulates deviations of sample values from a target mean $\mu_0$. Like EWMA, CUSUM is exceptionally effective for detecting small shifts ($0.5\sigma$ to $1.5\sigma$).

Tabular CUSUM Formulation

The standard approach in modern quality software is the Tabular (or Two-Sided) CUSUM. It maintains two cumulative statistics:

• $C_i^+$: Accumulates positive shifts above target (Upper CUSUM).
• $C_i^-$: Accumulates negative shifts below target (Lower CUSUM).

$$C_i^+ = \max \left( 0, x_i - (\mu_0 + k) + C_{i-1}^+ \right)$$ $$C_i^- = \max \left( 0, (\mu_0 - k) - x_i + C_{i-1}^- \right)$$

Where $C_0^+ = C_0^- = 0$.

CUSUM Parameters

• Reference Value ($k$): Also called the slack value or allowance. It is set midway between the target mean $\mu_0$ and the out-of-control mean $\mu_1 = \mu_0 + \delta \sigma$: $$k = \frac{\delta}{2} \sigma = \frac{|\mu_1 - \mu_0|}{2}$$ For detecting a $1.0\sigma$ shift ($\delta = 1.0$), $k = 0.5\sigma$.
• Decision Limit ($h$): The threshold distance beyond which an out-of-control signal is triggered. An out-of-control signal occurs if $C_i^+ > H$ or $C_i^- > H$, where $H = h \sigma$. Typically, $h$ is chosen as $4$ or $5$ (meaning $H = 4\sigma$ or $5\sigma$). Setting $h = 4.77$ with $k = 0.5$ provides an in-control Average Run Length $ARL_0 \approx 370$.

    CUSUM Signal Threshold H = h*σ
    ─────────────────────────────── Threshold H (Out of Control!)
                 /  \
                /    \   <- C_i+ Accumulation
    ───────────/──────\──────────── Baseline (0)

V-Mask Geometry

Historically, CUSUM charts utilized a graphic alignment tool called a V-Mask placed over the cumulative sum plot $S_i = \sum_{j=1}^i (x_j - \mu_0)$. The V-mask origin is placed on the latest point; if previous data points fall outside the V-mask arms (defined by origin distance $d$ and mask angle $\theta$), the process is out of control. Tabular CUSUM is mathematically equivalent and preferred for digital implementation.

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3. Short-Run SPC for Low-Volume Manufacturing

Standard SPC charts require $20$ to $25$ sub-groups of homogenous production to establish reliable control limits. In high-mix, low-volume environments (job shops, custom machining), batch sizes may be as small as $5$ to $50$ units, rendering traditional charts ineffective.

Standardized Z-Charts

Short-run SPC solves this by plotting standardized data ($Z$-scores) rather than raw measurements, allowing multiple part numbers with different means ($\mu_j$) and standard deviations ($\sigma_j$) to be monitored on a single chart.

$$Z_i = \frac{x_{ij} - \bar{x}_j}{s_j} \quad \text{or} \quad Z_i = \frac{x_{ij} - T_j}{\sigma_j}$$

Where $T_j$ is the target value for part type $j$. Because $Z_i$ has a mean of $0$ and a standard deviation of $1$, fixed control limits apply across all part numbers:

$$UCL = +3.0, \quad \text{Center Line} = 0, \quad LCL = -3.0$$

Difference-from-Nominal (DNOM) Charts

For short runs where the process variability $\sigma$ is constant across part numbers, but nominal targets $T_j$ differ, the Difference-from-Nominal ($DNOM$) transformation is applied:

$$y_{ij} = x_{ij} - T_j$$

The deviations $y_{ij}$ are plotted on a single $\bar{X}-R$ or $I-MR$ control chart with a center line of $0$.

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4. Multivariate SPC & Hotelling’s $T^2$

Modern products often require simultaneous control of $p$ correlated characteristics (e.g., length, width, and height of a precision component). Monitoring $p$ variables using $p$ independent univariate Shewhart charts introduces two critical failure modes:

1. Inflated Type I Error Rate: If each chart has $\alpha = 0.0027$ ($3\sigma$ limits), the overall probability of false alarm across $p$ independent charts is: $$\alpha_{\text{overall}} = 1 - (1 - \alpha)^p$$ For $p = 10$ variables, $\alpha_{\text{overall}} = 1 - (0.9973)^{10} \approx 0.0267$ (a false alarm every $37$ samples instead of $370$).
2. Distorted Control Boundaries: Independent limits form a rectangular control region, ignoring covariance between variables and missing out-of-control conditions that fall outside the true elliptical correlation boundary.

       Variable Y ^
                  |        / Elliptical Multivariate Boundary (T^2)
           Rectangular    /
          Univariate     /  * Out of Control in T^2 (Missed by Rectangular!)
          Limits  |  o  /
                  +----+
                  |    |
                  +----+---------> Variable X

Hotelling’s $T^2$ Statistic

For subgrouped data of $p$ correlated variables with sample size $n$, sample vector mean $\bar{\mathbf{x}}$, target mean vector $\boldsymbol{\mu}_0$, and sample covariance matrix $\mathbf{S}$:

$$T^2 = n (\bar{\mathbf{x}} - \boldsymbol{\mu}_0)' \mathbf{S}^{-1} (\bar{\mathbf{x}} - \boldsymbol{\mu}_0)$$

For individual observations ($n=1$):

$$T^2 = (\mathbf{x} - \boldsymbol{\mu}_0)' \mathbf{S}^{-1} (\mathbf{x} - \boldsymbol{\mu}_0)$$

Control Limits for Hotelling’s $T^2$

Unlike univariate charts, $T^2$ is a non-negative scalar distance measure; hence, $T^2$ charts have only an Upper Control Limit ($UCL$) and an $LCL = 0$.

For Phase II monitoring with $m$ preliminary samples of size $n$:

$$UCL = \frac{p(m-1)(n-1)}{mn - m - p + 1} F_{\alpha, p, mn - m - p + 1}$$

Diagnosing Out-of-Control Signals

When $T^2 > UCL$, the chart signals that the vector of variables has shifted, but it does not specify which variable(s) caused the shift. Quality engineers apply Mason-Young-Tracy (MYT) decomposition or Principal Component Analysis (PCA) to decompose $T^2$ into individual indicator components and isolate the root cause.

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5. Advanced SPC Method Comparison